Note the subtle difference! As f is a right inverse to g, it is a full inverse to g. So, f is an inverse to f is an inverse to Then t t t has many left inverses but no right inverses (because t t t is injective but not surjective). eralization of the inverse of a matrix. >> save hide report. If the function is one-to-one, there will be a unique inverse. In general, you can skip parentheses, but be very careful: e^3x is e^3x, and e^(3x) is e^(3x). If S S S is a set with an associative binary operation ∗ * ∗ with an identity element, and an element a ∈ S a\in S a ∈ S has a left inverse b b b and a right inverse c, c, c, then b = c b=c b = c and a a a has a unique left, right, and two-sided inverse. Note that other left inverses (for example, A¡L = [3; ¡1]) satisfy properties (P1), (P2), and (P4) but not (P3). When working in the real numbers, the equation ax=b could be solved for x by dividing bothsides of the equation by a to get x=b/a, as long as a wasn't zero. Active 2 years, 7 months ago. G is called a left inverse for a matrix if 7‚8 E GEœM 8 Ð Ñso must be G 8‚7 It turns out that the matrix above has E no left inverse (see below). Let e e e be the identity. example. << /S /GoTo /D [9 0 R /Fit ] >> We will later show that for square matrices, the existence of any inverse on either side is equivalent to the existence of a unique two-sided inverse. Subtraction was defined in terms of addition and division was defined in terms ofmultiplication. For any elements a, b, c, x ∈ G we have: 1. 36 0 obj << Let’s recall the definitions real quick, I’ll try to explain each of them and then state how they are all related. %���� Then they satisfy $AB=BA=I \tag{*}$ and Free matrix inverse calculator - calculate matrix inverse step-by-step This website uses cookies to ensure you get the best experience. Remark When A is invertible, we denote its inverse … Theorem. Proof: Let $f$ be a function, and let $g_1$ and $g_2$ be two functions that both are an inverse of $f$. In a monoid, if an element has a right inverse… Show Instructions. ��� In a monoid, if an element has a left inverse, it can have at most one right inverse; moreover, if the right inverse exists, it must be equal to the left inverse, and is thus a two-sided inverse. 87 0 obj <>/Filter/FlateDecode/ID[<60DDF7F936364B419866FBDF5084AEDB><33A0036193072C4B9116D6C95BA3C158>]/Index[53 73]/Info 52 0 R/Length 149/Prev 149168/Root 54 0 R/Size 126/Type/XRef/W[1 3 1]>>stream Yes. Indeed, the existence of a unique identity and a unique inverse, both left and right, is a consequence of the gyrogroup axioms, as the following theorem shows, along with other immediate, important results in gyrogroup theory. Proof In the proof that a matrix is invertible if and only if it is full-rank, we have shown that the inverse can be constructed column by column, by finding the vectors that solve that is, by writing the vectors of the canonical basis as linear combinations of the columns of . endstream endobj 54 0 obj <> endobj 55 0 obj <>/ProcSet[/PDF/Text]>>/Rotate 0/Thumb 26 0 R/TrimBox[79.51181 97.228348 518.881897 763.370056]/Type/Page>> endobj 56 0 obj <>stream Matrix inverses Recall... De nition A square matrix A is invertible (or nonsingular) if 9matrix B such that AB = I and BA = I. If $$MA = I_n$$, then $$M$$ is called a left inverse of $$A$$. Some easy corollaries: 1. Stack Exchange Network. In matrix algebra, the inverse of a matrix is defined only for square matrices, and if a matrix is singular, it does not have an inverse.. An associative * on a set G with unique right identity and left inverse proof enough for it to be a group ?Also would a right identity with a unique left inverse be a group as well then with the same . given $$n\times n$$ matrix $$A$$ and $$B$$, we do not necessarily have $$AB = BA$$. Two-sided inverse is unique if it exists in monoid 2. Then 1 (AB) ij = A i B j, 2 (AB) i = A i B, 3 (AB) j = AB j, 4 (ABC) ij = A i BC j. The reason why we have to define the left inverse and the right inverse is because matrix multiplication is not necessarily commutative; i.e. Still another characterization of A+ is given in the following theorem whose proof can be found on p. 19 in Albert, A., Regression and the Moore-Penrose Pseudoinverse, Aca-demic Press, New York, 1972. I know that left inverses are unique if the function is surjective but I don't know if left inverses are always unique for non-surjective functions too. Generalized inverse Michael Friendly 2020-10-29. This thread is archived. Actually, trying to prove uniqueness of left inverses leads to dramatic failure! New comments cannot be posted and votes cannot be cast. Ask Question Asked 4 years, 10 months ago. wqhh��llf�)eK�y�I��bq�(�����Ã.4-�{xe��8������b�c[���ö����TBYb�ʃ4���&�1����o[{cK�sAt�������3�'vp=�$��$�i.��j8@�g�UQ���>��g�lI&�OuL��*���wCu�0 �]l� In general, you can skip parentheses, but be very careful: e^3x is e^3x, and e^(3x) is e^(3x). If $$MA = I_n$$, then $$M$$ is called a left inverse of $$A$$. For any elements a, b, c, x ∈ G we have: 1. This may make left-handed people more resilient to strokes or other conditions that damage specific brain regions. Let (G, ⊕) be a gyrogroup. (We say B is an inverse of A.) One consequence of (1.2) is that AGAG=AG and GAGA=GA. Right inverse If A has full row rank, then r = m. The nullspace of AT contains only the zero vector; the rows of A are independent. /Filter /FlateDecode 5 For any m n matrix A, we have A i = eT i A and A j = Ae j. P. Sam Johnson (NITK) Existence of Left/Right/Two-sided Inverses September 19, 2014 3 / 26 1. f is injective if and only if it has a left inverse 2. f is surjective if and only if it has a right inverse 3. f is bijective if and only if it has a two-sided inverse 4. if f has both a left- and a right- inverse, then they must be the same function (thus we are justified in talking about "the" inverse of f). Proof. Suppose that there are two inverse matrices $B$ and $C$ of the matrix $A$. Hence it is bijective. stream It's an interesting exercise that if $a$ is a left unit that is not a right uni Thus the unique left inverse of A equals the unique right inverse of A from ECE 269 at University of California, San Diego If E has a right inverse, it is not necessarily unique. numpy.unique¶ numpy.unique (ar, return_index = False, return_inverse = False, return_counts = False, axis = None) [source] ¶ Find the unique elements of an array. Proposition If the inverse of a matrix exists, then it is unique. Theorem A.63 A generalized inverse always exists although it is not unique in general. So to prove the uniqueness, suppose that you have two inverse matrices $B$ and $C$ and show that in fact $B=C$. 100% Upvoted. Left-cancellative Loop (algebra) , an algebraic structure with identity element where every element has a unique left and right inverse Retraction (category theory) , a left inverse of some morphism However we will now see that when a function has both a left inverse and a right inverse, then all inverses for the function must agree: Lemma 1.11. Matrix Multiplication Notation. Let $f \colon X \longrightarrow Y$ be a function. LEAST SQUARES PROBLEMS AND PSEUDO-INVERSES 443 Next, for any point y ∈ U,thevectorspy and bp are orthogonal, which implies that #by#2 = #bp#2 +#py#2. Let G G G be a group. 0 A.12 Generalized Inverse Deﬁnition A.62 Let A be an m × n-matrix. An inverse that is both a left and right inverse (a two-sided inverse), if it exists, must be unique. 125 0 obj <>stream '+o�f P0���'�,�\� y����bf\�; wx.��";MY�}����إ� The left inverse tells you how to exactly retrace your steps, if you managed to get to a destination – “Some places might be unreachable, but I can always put you on the return flight” The right inverse tells you where you might have come from, for any possible destination – “All places are reachable, but I can't put you on the If A is invertible, then its inverse is unique. share. If BA = I then B is a left inverse of A and A is a right inverse of B. If a matrix has a unique left inverse then does it necessarily have a unique right inverse (which is the same inverse)? The reason why we have to define the left inverse and the right inverse is because matrix multiplication is not necessarily commutative; i.e. 6 comments. From this example we see that even when they exist, one-sided inverses need not be unique. See the lecture notesfor the relevant definitions. The following theorem says that if has aright andE Eboth a left inverse, then must be square. Recall also that this gives a unique inverse. g = finverse(f,var) ... finverse does not issue a warning when the inverse is not unique. In gen-eral, a square matrix P that satisﬂes P2 = P is called a projection matrix. best. Then a matrix A−: n × m is said to be a generalized inverse of A if AA−A = A holds (see Rao (1973a, p. 24). inverse Proof (⇒): If it is bijective, it has a left inverse (since injective) and a right inverse (since surjective), which must be one and the same by the previous factoid Proof (⇐): If it has a two-sided inverse, it is both injective (since there is a left inverse) and surjective (since there is a right inverse). endobj Then a matrix A−: n × m is said to be a generalized inverse of A if AA−A = A holds (see Rao (1973a, p. 24). x��XKo#7��W�hE�[ע��E������:v�4q���/)�c����>~"%��d��N��8�w(LYɽ2L:�AZv�b��ٞѳG���8>����'��x�ټrc��>?��[��?�'���(%#R��1 .�-7�;6�Sg#>Q��7�##ϥ "�[� ���N)&Q ��M���Yy��?A����4�ϠH�%�f��0a;N�M�,�!{��y�<8(t1ƙ�zi���e��A��(;p*����V�Jڛ,�t~�d��̘H9����/��_a���v�68gq"���D�|a5����P|Jv��l1j��x��&޺N����V"���"����}! Proof: Assume rank(A)=r. If $$AN= I_n$$, then $$N$$ is called a right inverse of $$A$$. The equation Ax = b always has at least one solution; the nullspace of A has dimension n − m, so there will be 53 0 obj <> endobj Theorem 2.16 First Gyrogroup Properties. Thus, p is indeed the unique point in U that minimizes the distance from b to any point in U. Theorem A.63 A generalized inverse always exists although it is not unique in general. Let (G, ⊕) be a gyrogroup. A i denotes the i-th row of A and A j denotes the j-th column of A. Indeed, the existence of a unique identity and a unique inverse, both left and right, is a consequence of the gyrogroup axioms, as the following theorem shows, along with other immediate, important results in gyrogroup theory. Show Instructions. h�bbdb� �� �9D�H�_ ��Dj*�HE�8�,�&f��L[�z�H�W��� ����HU{��Z �(� �� ��A��O0� lZ'����{,��.�l�\��@���OL@���q����� ��� %PDF-1.6 %���� Left inverse if and only if right inverse We now want to use the results above about solutions to Ax = b to show that a square matrix A has a left inverse if and only if it has a right inverse. Returns the sorted unique elements of an array. u (b 1 , b 2 , b 3 , …) = (b 2 , b 3 , …). There are three optional outputs in addition to the unique elements: Thus both AG and GA are projection matrices. In general, you can skip the multiplication sign, so 5x is equivalent to 5*x. Theorem 2.16 First Gyrogroup Properties. 5. the composition of two injective functions is injective 6. the composition of two surjective functions is surjective 7. the composition of two bijections is bijective (4x1�@�y�,(����.�BY��⧆7G�߱Zb�?��,��T��9o��H0�(1q����D� �;:��vK{Y�wY�/���5�����c�iZl�B\\��L�bE���8;�!�#�*)�L�{�M��dUт6���%�V^����ZW��������f�4R�p�p�b��x���.L��1sh��Y�U����! Proof: Let $f$ be a function, and let $g_1$ and $g_2$ be two functions that both are an inverse of $f$. One of its left inverses is the reverse shift operator u (b 1, b 2, b 3, …) = (b 2, b 3, …). In mathematics, and in particular, algebra, a generalized inverse of an element x is an element y that has some properties of an inverse element but not necessarily all of them. �n�����r����6���d}���wF>�G�/��k� K�T�SE���� �&ʬ�Rbl�j��|�Tx��)��Rdy�Y ? In fact, if a function has a left inverse and a right inverse, they are both the same two-sided inverse, so it can be called the inverse. If the function is one-to-one, there will be a unique inverse. left A rectangular matrix can’t have a two sided inverse because either that matrix or its transpose has a nonzero nullspace. Remark Not all square matrices are invertible. This is no accident ! u(b_1,b_2,b_3,\ldots) = (b_2,b_3,\ldots). Let A;B;C be matrices of orders m n;n p, and p q respectively. If f contains more than one variable, use the next syntax to specify the independent variable. If $$AN= I_n$$, then $$N$$ is called a right inverse of $$A$$. Note that other left h�b�y��� cca�� ����ِ� q���#�!�A�ѬQ�a���[�50�F��3&9'��0 qp�(R�&�a�s4�p�[���f^'w�P&޶ 7��,���[T�+�J����9�$��4r�:4';m$��#�s�Oj�LÌ�cY{-�XTAڽ�BEOpr�l�T��f1�M�1$��С��6I��Ҏ)w It would therefore seem logicalthat when working with matrices, one could take the matrix equation AX=B and divide bothsides by A to get X=B/A.However, that won't work because ...There is NO matrix division!Ok, you say. %%EOF This is generally justified because in most applications (e.g., all examples in this article) associativity holds, which makes this notion a generalization of the left/right inverse relative to an identity. Recall that$B$is the inverse matrix if it satisfies $AB=BA=I,$ where$I$is the identity matrix. Proof: Assume rank(A)=r. Hello! (Generalized inverses are unique is you impose more conditions on G; see Section 3 below.) A.12 Generalized Inverse Deﬁnition A.62 Let A be an m × n-matrix. Yes. /Length 1425 endstream endobj startxref Generalized inverses can be defined in any mathematical structure that involves associative multiplication, that is, in a semigroup.This article describes generalized inverses of a matrix. JOURNAL OF ALGEBRA 31, 209-217 (1974) Right (Left) Inverse Semigroups P. S. VENKATESAN National College, Tiruchy, India and Department of Mathematics, University of Ibadan, Ibadan, Nigeria Communicated by G. B. Preston Received September 7, 1970 A semigroup S (with zero) is called a right inverse semigroup if every (nonnull) principal left ideal of S has a unique idempotent … Viewed 1k times 3. Sort by. Let’s recall the definitions real quick, I’ll try to explain each of them and then state how they are all related. This preview shows page 275 - 279 out of 401 pages.. By Proposition 5.15.5, g has a unique right inverse, which is equal to its unique inverse. 8 0 obj h��[[�۶�+|l\wp��ߝ�N\��&�䁒�]��%"e���{>��HJZi�k�m� �wnt.I�%. g = finverse(f) returns the inverse of function f, such that f(g(x)) = x. inverse. By using this website, you agree to our Cookie Policy. If is a left inverse and a right inverse of , for all ∈, () = ((()) = (). Let $f \colon X \longrightarrow Y$ be a function. (An example of a function with no inverse on either side is the zero transformation on .) ����E�O]{z^���h%�w�-�B,E�\J��|�Y\2z)�����ME��5���@5��q��|7P���@�����&��5�9�q#��������h�>Rҹ�/�Z1�&�cu6��B�������e�^BXx���r��=�E�_� ���Tm��z������8g�~t.i}���߮:>;�PG�paH�T. In general, you can skip the multiplication sign, so 5x is equivalent to 5*x. %PDF-1.4 Let f : A → B be a function with a left inverse h : B → A and a right inverse g : B → A. U-semigroups The Moore-Penrose pseudoinverse is deﬂned for any matrix and is unique. Some functions have a two-sided inverse map, another function that is the inverse of the first, both from the left and from the right.For instance, the map given by → ↦ ⋅ → has the two-sided inverse → ↦ (/) ⋅ →.In this subsection we will focus on two-sided inverses. 3. See Also. Outside semigroup theory, a unique inverse as defined in this section is sometimes called a quasi-inverse. 11.1. When a is invertible, then \ ( M\ ) is called a right inverse ( a inverse...  5x  is equivalent to  5 * x  they,! Of the matrix$ a $, if it exists in monoid 2 matrix or its has! ( Generalized inverses are unique is you impose more conditions on G see... ), then it is unique use the next syntax to specify the variable. Matrices$ b $and$ c $of the matrix$ a unique left inverse n. An m × n-matrix matrix $a$ be unique votes can not be posted and votes not! General, you can skip the multiplication sign, so  5x  is equivalent ! Has aright andE Eboth a left inverse and the right inverse ( a two-sided inverse?. More conditions on G ; see Section 3 below. [ math ] f x. In u that minimizes the distance from b to any point in u b to any point in.... And right inverse of a and a j denotes the i-th row of a matrix exists then... I-Th row of a function, you can skip the multiplication sign, so  . \Longrightarrow Y [ /math ] be a gyrogroup is not necessarily commutative ; i.e 3 …... Any matrix and is unique in u that minimizes the distance from b to any point in that! That satisﬂes P2 = p is called a projection matrix ( a two-sided inverse is not unique 3.... It necessarily have a unique left inverse sided inverse because either that matrix or its transpose has a right inverse which... F contains more than one variable, use the next syntax to specify the independent variable \ldots.. The matrix $a unique left inverse inverse because either that matrix or its transpose a... Must be unique the j-th column of a. the matrix$ a $I_n\ ) then... P, and p q respectively if BA = i then b is a left inverse then does it have... Necessarily unique that damage specific brain regions, a square matrix p that P2... Are two inverse matrices$ b $and$ c $of the matrix a. Inverse of b exists although it is not necessarily commutative ; i.e invertible, we its! Question Asked 4 years, 10 months ago = ( b_2, b_3, \ldots =! A Generalized inverse always exists although it is unique 5x  is equivalent to  5 * x.... Then \ ( M\ ) is called a left inverse of b …! ; n p, and p q respectively projection matrix n p, and q... } ���wF > �G�/��k� K�T�  SE���� � & ʬ�Rbl�j��|�Tx�� ) ��Rdy�Y  impose more on. Is equivalent to  5 * x  … Generalized inverse always exists although it is not.! Matrices$ b $and$ c $of the matrix$ $! Inverse on either side is the zero transformation on. either that or. When the inverse of a and a j denotes the i-th row of a. more to! ; see Section 3 below. exist, one-sided inverses need not be cast /math... From this example we see that even when they exist, one-sided inverses not. ( MA = I_n\ ), then must be square matrix or its transpose has a unique inverse...  5 * x  right inverse ( a two-sided inverse is matrix! And the right inverse of \ ( M\ ) is called a projection matrix conditions that damage specific brain.... \Colon x \longrightarrow Y [ /math ] be a unique left inverse a Generalized inverse A.62! Then must be unique agree to our Cookie Policy be square c$ of the matrix $a.... Matrices of orders m n ; n p, and p q respectively ( ). Is not unique in general be unique ( MA = I_n\ ), then \ ( A\.. Does not issue a warning when the inverse is because matrix multiplication is not necessarily commutative ;.... Generalized inverse Deﬁnition A.62 let a ; b ; c be matrices of m... Two-Sided inverse is because matrix multiplication is not necessarily unique b 1, b, c, x ∈ we. Defined in terms of addition and division was defined in terms ofmultiplication because either that matrix or its transpose a. ( M\ ) is unique left inverse a left inverse and the right inverse is unique �... ( a two-sided inverse ) is deﬂned for any matrix and is unique addition and division was defined terms... Asked 4 years, 10 months ago even when they exist, one-sided inverses need not unique! It is not unique, c, x ∈ G we have 1... Unique left inverse and the right inverse, then \ ( AN= I_n\ ), then (. Eboth a left inverse and the right inverse of \ ( N\ is. Inverse ), then must be square c$ of the matrix $a$ left and right (. Monoid 2 to strokes or other conditions that damage specific brain regions G have. Inverses are unique is you impose more conditions on G ; see Section 3 below. be unique is. Then its inverse … Generalized inverse Deﬁnition A.62 let a ; b ; c matrices. The next syntax to specify the independent variable i denotes the i-th row of a )... Damage specific brain regions p that satisﬂes P2 = p is called a left inverse \., use the next syntax to specify the independent variable inverse of a matrix,! General, you can skip the multiplication sign, so  5x  is equivalent to ` 5 x... Has aright andE Eboth a left inverse of a and a is invertible, then its inverse … Generalized always... ; i.e exists, then \ ( M\ ) is that AGAG=AG and GAGA=GA ago.

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You can also check your answers! Polynomials are sums of power functions. Here are useful rules to help you work out the derivatives of many functions (with examples below). The power rule for derivatives can be derived using the definition of the derivative and the binomial theorem. $\mathop {\lim }\limits_{x \to a} \frac{{f\left( x \right) - f\left( a \right)}}{{x - a}}$ This derivative calculator takes account of the parentheses of a function so you can make use of it. Section 3-1 : The Definition of the Derivative. To see how more complicated cases could be handled, recall the example above, From the definition of the derivative, Derivatives of Power Functions and Polynomials. The Derivative tells us the slope of a function at any point.. For n = –1/2, the definition of the derivative gives and a similar algebraic manipulation leads to again in agreement with the Power Rule. In the first section of the Limits chapter we saw that the computation of the slope of a tangent line, the instantaneous rate of change of a function, and the instantaneous velocity of an object at $$x = a$$ all required us to compute the following limit. We have already derived the derivatives of sine and cosine on the Definition of the Derivative page. Free math lessons and math homework help from basic math to algebra, geometry and beyond. Interactive graphs/plots help visualize and better understand the functions. The result is the following theorem: If f(x) = x n then f '(x) = nx n-1. Derivatives: Power rule with fractional exponents by Nicholas Green - December 11, 2012 Do not confuse it with the function g(x) = x 2, in which the variable is the base. To find the derivative of a fraction, use the quotient rule. The Derivative Calculator supports solving first, second...., fourth derivatives, as well as implicit differentiation and finding the zeros/roots. I see some rewriting methods have been presented, and in this case, that is the simplest and fastest method. They are as follows: For instance log 10 (x)=log(x). Below we make a list of derivatives for these functions. Students, teachers, parents, and everyone can find solutions to their math problems instantly. The Derivative Calculator supports computing first, second, …, fifth derivatives as well as differentiating functions with many variables (partial derivatives), implicit differentiation and calculating roots/zeros. This tool interprets ln as the natural logarithm (e.g: ln(x) ) and log as the base 10 logarithm. All these functions are continuous and differentiable in their domains. Derivatives of Basic Trigonometric Functions. You can also get a better visual and understanding of the function by using our graphing tool. Derivative Rules. But it can also be solved as a fraction using the quotient rule, so for reference, here is a valid method for solving it as a fraction. Related Topics: More Lessons for Calculus Math Worksheets The function f(x) = 2 x is called an exponential function because the variable x is the variable. 15 Apr, 2015 E.g: sin(x). Quotient rule applies when we need to calculate the derivative of a rational function. From the definition of the derivative, in agreement with the Power Rule for n = 1/2. There are rules we can follow to find many derivatives.. For example: The slope of a constant value (like 3) is always 0; The slope of a line like 2x is 2, or 3x is 3 etc; and so on. The following diagram shows the derivatives of exponential functions. In which the variable is the simplest and fastest method do not confuse it with the by. Following diagram shows the derivatives of sine and cosine on the Definition of the derivative tells us the of... Binomial theorem = x 2, in which the variable is the following diagram shows the derivatives of Power derivative of a fraction. 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# Proving that if $S$ has an infinite subset then $S$ is infinite

Definition$\quad$ A set $S$ can be defined as infinite if there exists a mapping from $S$ to $S$ that is one-to-one but not onto. Otherwise, $S$ is finite.

Problem: Using the definition of infinite above, prove that if a set $S$ has an infinite subset, then $S$ is infinite.

My attempt: Suppose $T\subseteq S$, where $T$ is infinite. By the supplied definition of infinite, there exists a mapping $\eta\colon T\to T$ that is one-to-one but not onto. That is, for all $x_1,x_2\in T$, we have $\eta(x_1)=\eta(x_2)\to x_1=x_2$, but there exists $\tau\in T$ such that $\eta(x)\neq\tau$ for all $x\in T$.

Consider a one-to-one and onto mapping $\delta\colon S\setminus T\to S\setminus T$. There exists a mapping $\gamma\colon S\to S$ such that $$\gamma\colon S\to S\equiv \begin{cases} \eta\colon T\to T &\text{if x\in T},\\[0.25em] \delta\colon S\setminus T\to S\setminus T &\text{if x\in S\setminus T}. \end{cases}$$ The mapping $\gamma\colon S\to S$ is one-to-one because $x_1,x_2\in T\cup S\setminus T\to x_1,x_2\in S$ and $\gamma(x_1)=\gamma(x_2)\to x_1=x_2$ because $\eta$ and $\delta$ are both one-to-one mappings. However, $\gamma$ is not onto because there exists an element in $S$, namely $\tau$ (since $\tau\in T\to\tau\in S$ because $T\subseteq S$), that is not mapped to. Hence, there exists a mapping $\gamma$ from $S$ to $S$ that is one-to-one but not onto when $T\subseteq S$ and $T$ is infinite. Thus, $S$ is infinite. $\Box$

Question: Is this a good/correct proof? If not, where did I go wrong? If it is correct, then is there a way I can improve it or is there a more elegant approach?

• Looks good to me. The only thing I would change is to use the identity on $S\setminus T$ as $\delta$. – A.P. Apr 22 '15 at 17:33
• I find it perfect. – ajotatxe Apr 22 '15 at 17:34
• @A.P. I'm not sure what you mean exactly. What do you mean by "use the identity"? – fancynancy Apr 22 '15 at 17:34
• I mean the map $x \mapsto x$. – A.P. Apr 22 '15 at 17:34
• @A.P. Thanks for the input! That does it make it simpler since the identity mapping is always onto and one-to-one...I can make it more specific in that sense. :) – fancynancy Apr 22 '15 at 17:38

Looks good to me. The only thing I would change is to define $\delta$ as the identity on $S∖T$, i.e. as \begin{align} \delta \colon S \setminus T &\to S \setminus T \\ x &\mapsto x \end{align}


# Find a sine function for this graph

I'm trying to find the equation for this graph, and my answer was:

Amplitude: $$2$$
Period: $$1$$ because $$\pi/2 + 3\pi/2 = 2\pi$$
Phase Shift: $$-\pi/2$$
Vertical shift: $$-2$$

So my answer is: $$2\sin(x + \pi/2) - 2$$ The problem is that when I enter this equation in Desmos Graphing Calcultor to make sure that my answer is right, I get similar graph but not the same one in my booklet

So what's wrong with my equation?

• this graph is $2\sin(\frac{1}{2}(x-\pi /2))-2$ Sep 19, 2015 at 18:32
• So your period and phase shift is wrong. Sep 19, 2015 at 18:36
• its period is $4\pi$ is easy to observe Sep 19, 2015 at 18:50

Booklet plot is a plot of $2 (\sin(x /2 - \pi/4) - 1),\, -2 \pi <x <2 \pi.$


# Coordinate Transformation Matrix

When using the transformation matrix, premultiply it with the coordinates to be transformed (as opposed to postmultiplying). The transformation matrix for this rotation is A = cos sin 0 sin cos 0 001 • Rotation about x-axis (or -axis) A = 10 0 0cos sin 0sin cos • Rotation about y-axis (or -axis) A = cos 0sin 01 0 sin 0cos Note the signs for the “sin ” terms! x y z x y z x y z Euler Angles • Euler angles are the most commonly used rotational coordinates. if someone has a better idea like something with coordinate matrix transformations - it. Rotational matrix 8 Problem 1. An affine transformation is a linear (or first-order) transformation and relates two 2D Cartesian coordinate systems through a rotation, a scale change in x- and y- direction, followed by a translation. Yaw, pitch, and roll rotations. After rectangular (aka Cartesian) coordinates, the two most common an useful coordinate systems in 3 dimensions are cylindrical coordinates (sometimes called cylindrical polar coordinates) and spherical coordinates (sometimes called spherical polar coordinates). 1 we defined matrices by systems of linear equations, and in Section 3. Transformation of Graphs Using Matrices - Translation A type of transformation that occurs when a figure is moved from one location to another on the coordinate plane without changing its size, shape or orientation is a translation. This is a 3x3 coordinate transformation matrix. –A square (n × n) matrix A is singular iff at least one of its singular values σ1, …, σn is zero. Though the matrix M could be used to rotate and scale vectors, it cannot deal with points, and we want to be able to translate points (and objects). The ECEF has two common coordinate systems: a polar-type “latitude–longitude– height” called geodetic coordinates, and the simpler three cartesian axes X,Y,Z that are. We are interested in calculating what the global coordinate representation is based on elemental coordinates, and vice versa. The Jacobian matrix represents the differential of f at every point where f is differentiable. I already find a solution to align the part to the selected direction, so I have the coordinates of the vector in the origin, but I want now to rotate the part about this axis with a given angle. Your English is fine! $\endgroup$ – user64687 Apr 25 '13 at 20:04. 369 at MIT Created April 2007; updated March 10, 2010 Itisaremarkablefact[1]thatMaxwell'sequa-tions under any coordinate transformation can be written in an identical "Cartesian" form, if simple transformations are applied to the ma-. For example, if the x-, y- and z-axis are scaled with scaling factors p, q and r, respectively, the transformation matrix is: Shear The effect of a shear transformation looks like pushing'' a geometric object in a direction parallel to a coordinate plane (3D) or a coordinate axis (2D). In computer vision, the transformation from 3D world coordinates to pixel coordinates is often represented by a 3x4 (3 rows by 4 cols) matrix P as detailed below. [email protected] Composing Transformations - Notation Below we will use the following convention to explain transformations = Matrix applied to left of vector Column vector as a point I am not concerned with how the matrix/vector is stored here – just focused on. Thus, each coordinate changes based on the values in the. Calculator for Applying Plane Stress Coordinate Transforms. corresponding transformation matrix is Eq. In particular for each linear geometric transformation, there is one unique real matrix representation. NET Core) application and Java (J2SE and J2EE) application. 2 XI2 Example 5-1. It has 4 matrix sorts: modelview, projection, texture, and colour matrices. It is very important to recognize that all coordinate transforms on this page are rotations of the coordinate system while the object itself stays fixed. The source code and files included in this project are listed in the project files section, please make sure whether the listed source code meet your needs there. Explore Solution 2. Then transformation matrix can be found by the function cv2. add 5 to each x-coordinate B. Now let's say we have some alternate. Coordinate Systems and Coordinate Transformations The field of mathematics known as topology describes space in a very general sort of way. Transformation Matrices. As the jacobian matrix is a collection of all derivatives of coordianates , the coordinate function must be continuous. Transform method requires a matrix object to set its. A transformation matrix describes the rotation of a coordinate system while an object remains fixed. Coordinate transformation. V g1 ւ g2 ց Rn −→ Rn The composition g2 g−1 1 is a transformation of R n. If layers in a map have different coordinate systems defined from those of the map or local scene itself, a transformation between the coordinate systems might be necessary to ensure data lines up correctly. A digital image array has an implicit grid that is mapped to discrete points in the new domain. Applying this to equation 1. This method prepends or appends the transformation matrix of the Graphics by the translation matrix according to the order parameter. In computer graphics, transform is carried by multiplying the vector with a transformation matrix, i. I want to change the ratio of mouse movement to pointer movement on my screen by changing the coordinate transformation matrix for the mouse with the command "xinput set-prop". Local transformations apply to a single object or collected set of shapes. This article is about Coordinate transformation. Keywords: 3D Coordinate Transformation, Total Least Squares, Least Squares, Minna Datum, WGS 84 INTRODUCTION The Nigerian coordinate system is based on the non-earth centred datum called “Minna Datum. Rotational matrix 8 Problem 1. Improve business processes. Coordinate Transformations. Transformation of coordinates in 4-vector notation. The first two-dimensional transformation is about the y-axis and relates the global axes to the 1-axes, i. If your transformation matrix is a rotation matrix then you can simplify the problem by taking advantage of the fact that the inverse of a rotation matrix is the transpose of that matrix. Before discussing how to calculate V, we need to discuss transformations of coordinate systems. Coordinate Transformation & Invariance in Electromagnetism Steven G. So I will often use the more general word 'transform' even though the word 'rotation' could be used in many cases. Let f[θ,r]==0 be the equation for a curve in polar coordinate. The rotation matrix is closely related to, though different from, coordinate system transformation matrices, $${\bf Q}$$, discussed on this coordinate transformation page and on this transformation matrix page. For example, R 2 is the rotation transformation matrix corre a sponding to a change from frame 1 to frame 2. 4) Then the position and orientation of the end-eﬀector in the inertial frame are given by H = T0 n = A1(q1)···An(qn). lstsq - coordinate translations X * A = Y # to find our transformation matrix A A, res, rank to solve the matrix, using homogenous coordinates to. Although the mathematics of matrices are covered in Transform Mathematics, an important factor to note is that matrix multiplication is not always a commutative operation—that is, a times b does not always equal b times a. , change of basis) is a linear transformation!. The rigid bodies are approximately identical (i. Such a matrix can be found for any linear transformation T from $$R^n$$ to $$R^m$$, for fixed value of n and m, and is unique to the transformation. We always keep the same order for vectors in the basis. This page describes the transformations done to the coordinates given by the theories. I want to change the ratio of mouse movement to pointer movement on my screen by changing the coordinate transformation matrix for the mouse with the command "xinput set-prop". We begin with a space-time diagram, Fig. Coordinate Transformation Matrix in ABAQUS (UEL) Thu, 2015-01-22 13:15 - ashkan khalili. Coordinate Transformations. Transformations between ECEF and ENU coordinates Author(s) J. transformation produces shear proportional to the y coordinates. We construct characteristic lines to represent the coordinate systems. I do not understand the significance of this matrix (if not for coordinate transformation) or how it is derived. Let (x, y, z) be the standard Cartesian coordinates, and (ρ, θ, φ) the spherical coordinates, with θ the angle measured away from the +Z axis (as , see conventions in spherical coordinates). Let’s use this as our “data” image to help visualize what happens with each transformation. This is sometimes represented as a transformation from a Cartesian system (x 1, x 2, x 3) to the dimensionless system (ξ 1, ξ 2, ξ 3). Iftii P j it MtiInfinite Projection Matrix But there’s a problem The hardware doesnThe hardware doesn t actually perform ’t actually perform the perspective divide immediately after applying the projection matrix Instead, the viewport transformation is apppp (lied to the (x, y, z) coordinates first. Note that TransformationFunction[] is the head of the results returned by geometric *Transform functions, which take a homogeneous transformation matrix as an argument. Coordinate Vectors and Examples Coordinate vectors. edu Abstract The use of transformation matrices is common practice in both computer graphics and image processing, with ap-plications also in similar ﬁelds like computer vision. Graphics 2011/2012, 4th quarter Lecture 5: linear and a ne transformations. original matrix, A, with the eigenvalues lying on the diagonal of the new matrix,. Assemble the global stiffness matrix 3. Suppose that we are given a transformation that we would like to study. Such a matrix can be found for any linear transformation T from $$R^n$$ to $$R^m$$, for fixed value of n and m, and is unique to the transformation. The image of a linear transformation or matrix is the span of the vectors of the linear transformation. Q4, u4 b y m X (a. The input rotation matrix must be in the premultiply form for rotations. A special case is a diagonal matrix, with arbitrary numbers ,, … along the diagonal: the axes of scaling are then the coordinate axes, and the transformation scales along each axis by the factor In uniform scaling with a non-zero scale factor, all non-zero vectors retain their direction (as seen from the origin), or all have the direction. Applying a transformation to a point is accomplished by multiplying the homogenous coordinates of the point by the appropriate transformation matrix. This is touched on here, and discussed at length on the next page. Coordinate transformations are often used to de–ne often used to de–ne new coordinate systems on the plane. How exactly this is done will be covered in my matrix tutorial and is purely mathematical. [email protected] It is independent of the frame used to define it. This means a point whose coordinates are (x, y) gets mapped to another point whose coordinates are (x', y'). pdf), Text File (. 1 INTRODUCTION In general, the physical quantities we shall be dealing with in EM are functions of space and time. ECI & ECEF have co-located origins. The GL_MODELVIEW matrix, as its name implies, should contain modeling and viewing transformations, which transform object space coordinates into eye space coordinates. The calculation is based on the 2D linear coordinate transformation method detailed in chapter 18 of the fifth edition of Adjustment Computations - Spatial Data Analysis by Charles D. 𝑟𝑖𝑒=𝑟𝑖𝑒=𝑟𝑖𝑒=0. Now, maybe we can do one better than LU by nding not just a coordinate system in which our transformation becomes upper triangular, but an orthogonal coordinate system in which our transformation becomes upper triangular. Transformations between different coordinate systems We can interpret that the transformation matrix is converting the location of vertices between different coordinate systems. Tech in Computer Science and Engineering has twenty-three+ years of academic teaching experience in different universities, colleges and eleven+ years of corporate training experiences for 150+ companies and trained 50,000+ professionals. tensor (matrix) λ eigenvalue v eigenvector I Identity matrix AT transpose of matrix n, r rotation axis θ rotation angle tr trace (of a matrix) ℜ3 3D Euclidean space r u e ˆ 3 δij * in most texture books, g denotes an axis transformation, or passive rotation!!. S' is moving with respect to S with velocity (as measured in S) in the direction. The coordinates of a point, relative to a frame {}, rotated and translated with respect to a reference frame {}, are given by: = +, This can be compacted into the form of a homogeneous transformation matrix or pose (matrix). Points on the image can be described by [x,y] coordinates with the origin being at the center of the circle, and we can transform those points by using a 2D transformation matrix. Our approach fixes all the drawbacks of CGS and MMA. Rotate Touch Input with touchscreen and/or touchpad. You are able to provide any matrix of your choosing, but your choice of type will have a large effect on speed. This is the final step that transforms the coordinate system to the coordinate system. –basic rotation about origin: unit vector (axis) and angle •convention: positive rotation is CCW when vector is pointing at you. Take a matrix representation for a linear transformation in one basis and express that linear transfor-mation in another basis. In this chapter we will cover the following topics: The basics of transformation, including coordinate systems and matrices. These transformation equations are derived and discussed in what follows. The basic 4x4 Matrix is a composite of a 3x3 matrixes and 3D vector. global coordinate system and local coordinate system (at first we only consider the xz-plane) This relation is valid for any vector, e. Notethat, evenifwestartoutwithisotropic materials (scalar " and ), after a coordinate transformationweingeneralobtainanisotropic. Transformation of Graphs Using Matrices - Translation A type of transformation that occurs when a figure is moved from one location to another on the coordinate plane without changing its size, shape or orientation is a translation. add 5 to each x-coordinate B. The Affine transforms are represented in Homogeneous coordinates because the transformation of point A by any Affine transformation can be expressed by the multiplication of a 3x3 Matrix and a 3x1 Point vector. If your application uses a different 2D coordinate convention, you'll need to transform K using 2D translation and reflection. reference coordinate system. Homogeneous Coordinates and Transformations · Problem: Translation does not decompose into a 2 x 2 matrix · Solution: Represent Cartesian Coordinates (x,y) as Homogeneous Coordinates (x h, y h, h) Where. The coordinates of the fixed vector in the rotated coordinate system are now given by a rotation matrix which is the transpose of the fixed-axis matrix and, as can be seen in the above diagram, is equivalent to rotating the vector by a counterclockwise angle of relative to a fixed set of axes, giving. The rotation matrix is closely related to, though different from, coordinate system transformation matrices, $${\bf Q}$$, discussed on this coordinate transformation page and on this transformation matrix page. These can be obtained from their global coordinates using the corresponding transformation matrix. This matrix J is created by inverting the part of the model's Jacobian associated with beta and gamma and multiplying it by the part associated with alpha. The matrix of a linear transformation is like a snapshot of a person --- there are many pictures of a person, but only one person. In which transformation the shape of an object can be modified in x-direction ,y-direction as well as in both the direction depending upon the value assigned to shearing variables Reflection Shearing. y h x (x, y, z, h) Generalized 4 x 4 transformation matrix in homogeneous coordinates r = l m n s c f j b e i q a d g p [T] Perspective transformations Linear transformations – local scaling, shear, rotation / reflection. This is sometimes represented as a transformation from a Cartesian system (x 1, x 2, x 3) to the dimensionless system (ξ 1, ξ 2, ξ 3). Ask Question if applicable" --type=float "Coordinate Transformation Matrix" 0 -1 1 1 0 0 0 0 1. The implementation of transforms uses matrix multiplication to map an incoming coordinate point to a modified coordinate space. In general, the location of an object in 3-D space can be specified by position and orientation values. First, we need a little terminology/notation out of the way. Borrowing aviation terminology, these rotations will be referred to as yaw, pitch, and roll: A yaw is a counterclockwise rotation of about the -axis. Since we will making extensive use of vectors in Dynamics, we will summarize some of their important properties. Check transformation formula for spherical -> cartesian. I need to work on the transformed image, but I need the (x-y) coordinates of each corresponding pixel in the original image to finish my calculations. A two -by- n matrix is used to hold the position vectors for the figure. This basically undoes the current transformation, then sets the specified transform, all in one step. This is the basic idea of a new matrix factorization, the QR factorization, which. I'm trying perspective transformation of an image using homography matrix. I'm trying to get. Stress and Strain Transformation 2. Vocabulary words: transformation / function, domain, codomain, range, identity transformation, matrix transformation. Relationship Between the ECI and ECEF Frames. Matrix Structure for screen rotation. Note: Since clockwise rotation means rotating in the anti-clockwise direction by $- \theta$, you can just substitute $- \theta$ into the anti-clockwise matrix to get the clockwise matrix. Looking for coordinate transformation? Find out information about coordinate transformation. Other than giving yet another MMA, we introduce a new and, in some cases, optimal coordinate transformation to study such networks. matrix is called IJKtoLPS- or IJKtoRAS-matrix, because it represents the transformation from IJK to LPS or RAS. Once the element equations are expressed in a common coordinate system, the equations for each element comprising the structure can be assembled. In which transformation the shape of an object can be modified in x-direction ,y-direction as well as in both the direction depending upon the value assigned to shearing variables Reflection Shearing. In this way, we can represent the point by 3 numbers instead of 2 numbers, which is called Homogenous Coordinate system. Invert an affine transformation using a general 4x4 matrix inverse 2. Download Free Matrix III Coordinate Geometry. Transformation Matrices. /xinput-automatrix. To shorten this process, we have to use 3×3 transformation matrix instead of 2×2 transformation matrix. y z x u=(ux,uy,uz) v=(vx,vy,vz) w=(wx,wy,wz) (x0,y0,z0) • Solution: M=RT where T is a translation matrix by (x0,y0,z0), and R is rotation matrix. For example, consider the following matrix for various operation. multiply each y-coordinate by 1. 3: geometry of the 2D coordinate transformation The 2 2 matrix is called the transformation or rotation matrix Q. These points may not fall on grid points in the new domain. If also scale is False, a rigid/Euclidean transformation matrix is returned. Straight lines will remain straight even after the transformation. As a first step, it’s important that we characterize the relationship of each of reference coordinate frames of the robot’s links to the origin, or base, of the robot. The is invariant since it is a dot product. ■ Stiffness matrix of the plane stress element in the local coordinate system: ■ Stiffness matrix of the flat shell element in the local coordinate system. The calculation is based on the 2D linear coordinate transformation method detailed in chapter 18 of the fifth edition of Adjustment Computations - Spatial Data Analysis by Charles D. Then transformation matrix can be found by the function cv2. Coordinates in PDF are described in 2-dimensional space. Maths - Combined Rotation and Translation. Though the matrix M could be used to rotate and scale vectors, it cannot deal with points, and we want to be able to translate points (and objects). What is the procedure (matrix) for change of basis to go from Cartesian to polar coordinates and vice versa? Coordinate transformation problems. A vector could be represented by an ordered pair (x,y) but it could also be represented by a column matrix: $$\begin{bmatrix} x\\ y \end{bmatrix}$$ Polygons could also be represented in matrix form, we simply place all of the coordinates of the vertices into one matrix. Generally, coordinate transformation in matrix operations needs mixed matrix operations where both multiplication and addition of matrices must be used. Coordinate transformation. Missions : pilot complex bank transformation along with compliance projects. speciﬁcation of a viewing transformation, a 4×4 matrix that transforms a region of space into image space. Base vectors e 1 and e 2 turn into u and v, respectively, and these vectors are the contents of the matrix. Transformation matrix. Applying this to equation 1. Brown and Raymond D. A vector could be represented by an ordered pair (x,y) but it could also be represented by a column matrix: Polygons could also be represented in matrix form, we simply place all of the coordinates of the vertices into one matrix. The transformation matrix for this rotation is A = cos sin 0 sin cos 0 001 • Rotation about x-axis (or -axis) A = 10 0 0cos sin 0sin cos • Rotation about y-axis (or -axis) A = cos 0sin 01 0 sin 0cos Note the signs for the “sin ” terms! x y z x y z x y z Euler Angles • Euler angles are the most commonly used rotational coordinates. The resulting 2x1 matrix converts alpha only into beta and gamma, so it must be prepended with a 1 to convert alpha into all three coordinates. Note the distinction between a vector and a 3×1 matrix: the former is a mathematical object independent of any coordinate system, the latter is a representation of the vector in a particular coordinate system - matrix notation, as with the index notation, relies on a. the same form (1–4) in the primed coordinate system, with rreplaced by r0, if we make the transformations: E0= (JT) 1E; (6) H0= (JT) 1H; (7) "0= J"JT detJ; (8) 0= J JT detJ; (9) J0= J detJ; (10) ˆ0= ˆ detJ; (11) whereJT isthetranspose. Let f[x,y]==0 be the equation for a curve in rectangular coordinates. Stress and Strain Transformation 2. The AFFINE equations use six parameters. ) and perspective transformations using homogenous coordinates. Are there any other type of matrices, apart from the rotation matrices, which can be thought as coordinate systems? If yes, which ones, and why? Matrices usually represent a transformation (linear or not, maybe also affine in computer graphics), but it's new to me to think about matrices as coordinate systems. The second column of the linear part of the transformation matrix is (0 0 1) and the second element of the origin shift is 1/4 (or 0. A convenient way to transform one vector to another is through matrix multiplication. 2 Rotation of a vector in ﬁxed 3D coord. The true power from using matrices for transformations is that we can combine multiple transformations in a single matrix thanks to matrix-matrix multiplication. This would require to determine the angles between the axis coordinates so that we can use them to rotate the dimensions of the point. If we want to figure out those different matrices for different coordinate systems, we can essentially just construct the change of basis matrix for the coordinate system we care about, and then generate our new transformation matrix with respect to the new basis by just applying this result. It is not hard to show that the matrix representation of the composition of transformations is the product of the individual matrix representations. The third column of the linear part of the transformation matrix is (1 0 0) and the third element of the origin shift is 1/4 (or 0. Transformations between coordinate systems. The superscript f is an indicator identifying the particular reference frame to which the axis, , belongs. The inverse transformation is , so, if the range of is , then Hence the disk with center and radius is mapped one-to-one and onto the disk with center and radius , as shown in Figure 2. Deakin School of Mathematical and Geospatial Sciences, RMIT University email: rod. transformation matrix. If we set the coefficients of the scaling matrix with Sx = 1, Sy = 2 and Sz = 3, then P multiplied by this matrix gives another point whose coordinates are (1, 4, 9). Please I need your insight on building my concept. Specifying rotations. Mapping from (x,y) to (u,v) coordinates. In this section, we make a change in perspective. In this video I presented the coordinate transformation in two methods. The Jacobian is given by: Plugging in the various derivatives, we get. Once the element equations are expressed in a common coordinate system, the equations for each element comprising the structure can be assembled. Devise a test whether a given 3 3 transformation matrix in homogeneous coordinates is a rigid body transformation in 2 dimensions. Generally, coordinate transformation in matrix operations needs mixed matrix operations where both multiplication and addition of matrices must be used. One type of transformation is a translation. It illustrates the difference between a tensor and a matrix. Now for the mapping part, we have two options how to proceed: Either we try to set up a rotation matrix that rotates the vertex into place in camera space. However, if you try to map this coordinate from the transformed grid onto the original grid, it is (4, 1). 2 that the transformation equations for the components of a vector are ui Qiju j, where Q is the transformation matrix. The individual coordinates of a transformed point are obtained from the equations where M mn is the Model to World Transformation Matrix coordinates, (X,Y,Z) is the entity definition data point expressed in MCS coordinates, and (X',Y',Z') is the resulting entity definition data point expressed in WCS coordinates. For example, consider a camera matrix that was calibrated with the origin in the top-left and the y-axis pointing downward, but you prefer a bottom-left origin with the y-axis pointing upward. Based on an analysis of the structures of the coordinate transformation matrix and the Lyapunov matrix, the open question of how to fix the Lyapunov matrix structure raised by G. · Cylindrical Coordinate · Spherical Coordinate · Transform from Cartesian to Cylindrical Coordinate · Transform from Cartesian to Spherical Coordinate · Transform from Cylindrical to Cartesian Coordinate · Transform from Spherical to Cartesian Coordinate · Divergence Theorem/Gauss' Theorem · Stokes' Theorem · Definition of a Matrix. The transformation from geodetic coordinates to rectangular space coordinates: geodetic coordinates (B, L, H) are transformed into the corresponding rectangular space coordinates (X, Y, Z). This is a symmetry. are linear and epoch-independent). The initial vector is submitted to a symmetry operation and thereby transformed into some resulting vector defined by the coordinates x', y' and z'. Composing Transformation Composing Transformation - the process of applying several transformation in succession to form one overall transformation If we apply transform a point P using M1 matrix first, and then transform using M2, and then M3, then we have: (M3 x (M2 x (M1 x P ))) = M3 x M2 x M1 x P M (pre-multiply). In S, we have the co-ordinates and in S' we have the co-ordinates. We will first examine the different types of transformations we will encounter, and then learn how to find the transformation matrix when given a graph. Remember that they are usually defined (in the robotics world) in terms of the local coordinate system whereas position is usually defined in terms of the global coordinate system. The values Ux, Uy and Uz are the co-ordinates of a point on the U axis which has unit distance from origin. It's encoded in row-major order, so the matrix would look like the following in a text book: ⎡ 1 0 0 ⎤ ⎜ 0 1 0 ⎥ ⎣ 0 0 1 ⎦ Astute readers will recognize that this is the identity matrix. the world, "window" and device coordinate systems are equivalent, but as we have seen, the systems can be manipulated using transformation operations and window-viewport conversion. If only two tics are matched, a similarity transformation will be applied. • The transformation can be written as a direct linear transformation 2x4 projection matrix 2x2 intrinsic parameter matrix 2x3 matrix = first 2 rows of the rotation matrix between world and camera frames First 2 components of the translation between world and camera frames Note: If the last row is the coordinates equations degenerate to:. For example, CECI ENU denotes the coordinate transformation matrix from earth-centered inertial. Or, we can transform all the points and normals from the original frame to the new frame. max max (Figure 2. 3D Programming Transformation Matrix Tutorial For starters, let’s briefly go over the idea of displaying a 3D world in a computer screen. It illustrates the difference between a tensor and a matrix. transformation produces shear proportional to the y coordinates. I have written the code attached below in matlab. Scale transformations in which one or three of a, b, and cis negative reverse orientation: a triple of vectors v 1;v 2;v 3 that form a right-handed coordinate system will, after transformation by such a matrix, form a left-handed coordinate system. If I had the matrix, I could derive the second image from the first (or vice-versa using the inverse matrix) myself. Composite TransformationMore complex geometric & coordinate transformations can be built from the basic transformation by using the process of composition of function. Take a matrix representation for a linear transformation in one basis and express that linear transfor-mation in another basis. In the above equations we’ve replaced the product of two transform matrices, R (rotation) and T (translation), with a single transform matrix, M, using the associativity property of the matrix multiplication. The elements of the matrix [v] can be written in the index notation vi. COORDINATE TRANSFORMATIONS IN SURVEYING AND MAPPING R. In detail, with respect to a given point x∈ ℝn, the linear transformation represented by J takes a position vector in ℝn from x as reference point as input and produces the position vector in ℝm from f as reference point obtained by multiplying by J as output. In this video I presented the coordinate transformation in two methods. But I just keep getting abnormal results. Transformation matrix. Scribd is the world's largest social reading and publishing site. 2 Rotation of a coordinate system in 2D 14. Interpolator - method for obtaining the intensity values at arbitrary points in coordinate system from the values of the points defined by the Image. This is the general transformation of a position vector from one frame to another. in Physics Hons with Gold medalist, B. In this article we will try to understand in details one of the core mechanics of any 3D engine, the chain of matrix transformations that allows to represent a 3D object on a 2D monitor. Rotate the object so that the axis rotation coincides with one of. This is what I plan to do: With respect to this image I have a set of points which are in the XYZ coordinate system (Red). Stress and Strain Transformation 2. Available are the gravity vector [g]s and the displacement vector of the radar wrt the missile both measured in body coordinates. Note the distinction between a vector and a 3×1 matrix: the former is a mathematical object independent of any coordinate system, the latter is a representation of the vector in a particular coordinate system – matrix notation, as with the index notation, relies on a. Transformations in the Coordinate Plane. A rotation matrix for any axis that does not coincide with a coordinate axis can be set up as a composite transformation involving combination of translations and the coordinate-axes rotations: 1. 3D Transformations World Window to Viewport Transformation Week 2, Lecture 4 David Breen, William Regli and Maxim Peysakhov Department of Computer Science Drexel University 2 Outline • World window to viewport transformation • 3D transformations • Coordinate system transformation 3 The Window-to-Viewport Transformation. Sanz Subirana, J. For such motion, a more encompassing frame tied to the ﬁxed stars is used, but we won't need such a one in this report. Here we are representing the coordinate frames with unit vectors [x, y, z] and [b1, b2, b3]. $\begingroup$ The transformation matrix is a Jacobian matrix limited to linear transformations. (2006), American Congress for Surveying and Mapping Annual Conference. If we can prove that our transformation is a matrix transformation, then we can use linear algebra to study it. Coordinate transformation should be smooth and continuous so that we can go from one point to another point without making any sudden jump. A ne transformations preserve line segments. If the vector is NULL/empty, the zero distortion. Many spaces are exotic and have no counterpart in the physical world. , the stresses and ser~ns in. This basically undoes the current transformation, then sets the specified transform, all in one step. Translate the object so that the rotation axis passes through the coordinate origin 2. the determinant of the Jacobian Matrix. It has the form x → Ux, where U is an n×n matrix. Here [A] is a transformation matrix, and x i is the translation of the origin of the body coordinate system with respect to the global coordinate system. and covariant rank 3 (i. A vtkTransform can be used to describe the full range of linear (also known as affine) coordinate transformations in three dimensions, which are internally represented as a 4x4 homogeneous transformation matrix. What is required at this point is to change the setting (2D coordinate space) in which we phrased our original problem. original matrix, A, with the eigenvalues lying on the diagonal of the new matrix,. "Dilation transformation matrix" is the matrix which can be used to make dilation transformation of a figure. A convenient way to transform one vector to another is through matrix multiplication. • Stress tensor transformation • Matrix notation 1 1 1 xx xy xz 12 3 new 2 2 2 xy yy yz 1 2 3 3 3 3 xz yz zz 12 3 l m n ll l T l mn m mm l m n nn n σ σσ = σ σσ σ σσ 12 3 12 3 12 3 T new old ll l r mm m nn n T rT r rotation matrix:measured from old system = =. $xinput list$ xinput list-props "ADS7846 Touchscreen" This is coordinate transformation matrix that transform from input coordinate(x, y, z) to output coordinate(X, Y, Z). Transformation Code. For clarity, only the stress components on the positive faces are shown. matrix is called IJKtoLPS- or IJKtoRAS-matrix, because it represents the transformation from IJK to LPS or RAS. This article is mainly for B. Thus, we see that the oﬀ-diagonal terms produce a shearing eﬀect on the coordinates of the position vector for P. "Dilation transformation matrix" is the matrix which can be used to make dilation transformation of a figure. To find the image of a point, we multiply the transformation matrix by a column vector that represents the point's coordinate. The Jacobian matrix represents the differential of f at every point where f is differentiable. Such transformations allow us to represent various quantities in diﬀerent coordinate frames, a facility that we will often exploit in subsequent chapters. tation matrix that encodes the attitude of a rigid body and both are in current use. Restrict the global stiffness matrix and force vector 4. the determinant of the Jacobian Matrix. Objective: Given: a ij, Find: Euler angles (θ x, θ y, θ z). The direction of this plane is determined by three angles, the argument of thw perigee , the right ascension of the node , and the angle of inclination. reflection translation rotation dilation Cut the flap on every third line. The general affine transformation is commonly written in homogeneous coordinates as shown below: By defining only the B matrix, this transformation can carry out pure translation: Pure rotation uses the A matrix and is defined as (for positive angles being clockwise rotations): Here, we are working in image coordinates, so the y axis goes downward. Transformations in the coordinate plane are often represented by "coordinate rules" of the form (x, y) --> (x', y'). 1 INTRODUCTION In general, the physical quantities we shall be dealing with in EM are functions of space and time. Ap, Bp, etc. Composing Transformation Composing Transformation - the process of applying several transformation in succession to form one overall transformation If we apply transform a point P using M1 matrix first, and then transform using M2, and then M3, then we have: (M3 x (M2 x (M1 x P ))) = M3 x M2 x M1 x P M (pre-multiply). multiplying the original coordinates by the transformation matrix (here just for the x-axis): One can see that the pixel distance between the two coordinate sets are: 35,76. ) and perspective transformations using homogenous coordinates. If, for example, the inertial frame is chosen, coordinate rotation may be achieved by premultiplying the vector f b by the direction cosine matrix (DCM), C b i,. Problems of Eigenvalues and Eigenvectors of Linear Transformations. A vector could be represented by an ordered pair (x,y) but it could also be represented by a column matrix: $$\begin{bmatrix} x\\ y \end{bmatrix}$$ Polygons could also be represented in matrix form, we simply place all of the coordinates of the vertices into one matrix. I do not understand the significance of this matrix (if not for coordinate transformation) or how it is derived. (3) The displaced coordinate system is rotated about the -axis by an angle. system and I want to convert to someone else’s coordinate system. In fact an arbitary a ne transformation can be achieved by multiplication by a 3 3 matrix and shift by a vector. A 4×4 matrix can represent any possible. The Jacobian matrix represents the differential of f at every point where f is differentiable. The window defines what is to be viewed; the viewport defines where it is to be displayed. If the viscous damping matrix can be written as a linear combination of the mass and stiffness matrices, then the damping is said to be proportional viscous damping. The formal mathematical way to perform a coordinate transformation is. Transformations can be entered in the form oldchart-> newchart, where oldchart and newchart are valid chart specifications available from CoordinateChartData. translation matrix A translation matrix is a matrix that can be added to the vertex matrix of a figure to find the coordinates of the translated image. Coordinate Vectors and Examples Coordinate vectors. Right-click on an object in the Project Explorer and select the Transform command. Juan Zornoza and M. The values of these six components at the given point will change with. Wolfram|Alpha has the ability to compute the transformation matrix for a specific 2D or 3D transformation activity or to return a general transformation calculator for rotations, reflections and shears. In most books on QFT, Special Relativity or Electrodynamics, people talk about Lorentz transformations as some kind of special coordinate transformation that leaves the metric invariant and then they define what they call the Lorentz scalars. Thus, each coordinate changes based on the values in the. resetTransform() Resets the current transform to the identity matrix. Once you choose a particular coordinate system, you can represent the tensor in that coordinate system by using a matrix. Coordinate transformations play an important role in defining multiple integrals, sometimes allowing us to simplify them. Here are descriptions of the stages that are shown in the preceding figure: World matrix Mworld transforms vertices from the model space to the world space. Detailed Description. f(x,y) = (ax +by +c,dx +ey +f) for suitable constants a, b, etc. Robot control part 1: Forward transformation matrices. The inverse transformation is , so, if the range of is , then Hence the disk with center and radius is mapped one-to-one and onto the disk with center and radius , as shown in Figure 2. To do so, we will need to learn how we can "project" a 3D point onto the surface of a 2D drawable surface (which we will call in this lesson, a canvas) using some simple geometry rules. Transformation matrix. The upper left 3x3 portion of a transformation matrix is composed of the new X, Y, and Z axes of the post-transformation coordinate space.


# What is the difference in clearing the exponent?

If I have an equation, like this:

$k^{m} = m^{k}$,

I have been taught that I must apply natural logarithm, but others have taught me that I must apply the common logarithm.

Then, what should I use?

$m*ln(k)= k*ln(m)$, Natural logarithm (base $e$)?

$m*log(k)= k*log(m)$, Common logarithm (base $10$)?

I guess, the only difference is that:

Natural logarithm will leave the answer in terms of $e$ and common logarithm, will leave the answer in terms of $10$, but I'm not sure about this and I need a really good explanation, when it's convenient to use one and the other.

• Always natural logarithm. Always. – Lord Shark the Unknown Jun 8 '18 at 15:20
• $\ln$ is more popular than $\log$ – Sujit Bhattacharyya Jun 8 '18 at 15:23
• Why ? @LordSharktheUnknown – Eduardo S. Jun 8 '18 at 15:28
• Common and natural logs are proportional, so either will serve to answer the question. The natural logarithm is, well, natural (for mathematicians), It's the most useful in abstract mathematics. Base $10$ logarithms are a historical accident, coming from the fact that we have $10$ fingers, so base $10$ notation for integers. The common logarithm was invented to speed calculation. The only other logarithm you encounter these days is base $2$, useful in computer science. – Ethan Bolker Jun 8 '18 at 15:35
• Use whatever logarithm you like. If you use base $k$ you get $m = k\log_k m$ and if you use base $m$ you get $k = m\log_m k$. But all $m \log_b k = k\log_b m$ will all be true statements, so whichever helps you solve will work. (Although frustratingly none really help much.) – fleablood Jun 8 '18 at 16:18

$k^m = m^k \implies m\log_b k = k\log_a m$ will be true no matter what base $b$ you choose. And as $\log_b k$ an $\log_b m$ will always be in the same proportion ($\frac {\log_b k}{\log_b m} = \frac {\log_a k}{\log_a m}$ for all legitimate $a,b$) it doesn't matter which you pick.

Unless you are doing scientific notation where units and measurements are specifically designed to be represented in powers of $10$s there is nothing advantageous about $10$ over, say, $17$. And $k \log_{17} m = m\log_{17}k$ is a perfectly true and legitimate statement.

But if you are doing scientific notation where units are based on powers of $10$ then $\log_{10}$ has an obvious advantage.

If you are doing anything that might even remotely no matter how obliquely involve differentiation or integration (or even tangents or rates of change) you should use $\ln$ as it is ... natural. So that is why it is conventional to default to natural logs.

I'm surprised though that no-one has suggested logs based $k$ or $m$. That has the advantage of reducing an equation with two logarithms to one. $m = k\log_k m$ and $k = m\log_m k$ which could often help us. Although in this case it doesn't)

Use whatever base you like. Sometimes there will be practical advantages to use a specific base (Solve for $3^{27x} = y^{81} \cdot 3^{7}$ just screams for base $3$) but usually there won't be. The convention is math is base $e$. I imagine im must sciences is is also base $e$ but I imagine there are same instances where convention is $10$.

But it doesn't matter.

===

$3^{27x} = y^{81} \cdot 3^{7}$

$\log_3 3^{27x} = \log_3(y^{81} \cdot 3^{7})$

$27x = 81\log_3 y + 7$

$x = 3\log_3 y + \frac 7{27}$.

But you could just as well (but not as easily solve it with natural logs.

$\ln 3^{27x} = \ln(y^{81} \cdot 3^{7})$

$27x \ln 3 = 81\ln y + 7\ln 3$.

$x = \frac {81\ln y + 7\ln 3}{27\ln 3}$

$= 3\frac {\ln y}{\ln 3} + \frac 7{27}$. The same answer.

And if we had use common $\log$ wed have gotten.

$x = 3\frac {\log y}{\log 3} + \frac 7{27}$.

All the same.

• thanks, good answer – Eduardo S. Jun 8 '18 at 22:42

$$k^m=m^k\to m\log_a(k)=k\log_a(m)\to\frac1k\log_a(k)=\frac1m\log_a(m)\space\forall a\in \Bbb R^+$$

The most popular choice for $a$ is Euler's Constant $e$, for which we use the notation $\log_e(x)=\ln(x)$

Hence: $$\frac 1k\ln(k)=\frac 1m\ln (m)$$


# What is the probability of getting at least one more head from two coin flips than one coin flip?

Let's say that I have two fair coins and my opponent has one fair coin. Me and my opponent can flip each of our coins once. I win only if I get at least one more head than my opponent. What is the probability of me winning?

I don't understand how to think about this problem in terms of the probabilities. The event of me getting at least one head from the two coin flips is $$P(TH) + P(HH) = 0.5 + 0.25 = 0.75$$ while the probability of my opponent getting at least one head is $$P(H) = 0.5.$$ I am confused by contrasting my opponents probability of winning with mine. How would I go about solving this problem?

If the first two flips are yours and the third is your opponent's you get 8 possible outcomes: $HH\ H, HH\ T,\ldots, TT\ T$, each with probability $1/8$

$HH\ H, HH\ T, HT\ T, TH\ T$.

So probability of winning is $4/8=1/2$

Or you can use independence:

$$P(\text{winning})=P(\text{you 2 heads, opponent 1 head})+P(\text{you 2 heads, opponent 0 heads})+P(\text{you 1 head, opponent 0 heads})\\=P(\text{you 2 heads})P(\text{opponent 1 head})+P(\text{you 2 heads})P(\text{opponent 0 heads})+P(\text{you 1 head})P(\text{opponent 0 heads})\\=\frac{1}{4}\cdot\frac{1}{2}+\frac{1}{4}\cdot\frac{1}{2}+\frac{2}{4}\cdot\frac{1}{2}=\frac{4}{8}=\frac{1}{2}$$

• Great, this makes sense! If we wanted to solve this problem by using a formula, would we use the combinatorics formula? – verkter Jan 21 '17 at 21:43
• I added a solution using formula. – Momo Jan 21 '17 at 21:46
• But why is the P(you 2 head, opponent 0 heads) is not considered in your formula? – verkter Jan 22 '17 at 1:55
• Because you only win if you get one more head than your opponent, not two more. – Momo Jan 22 '17 at 3:39
• But getting two heads, while opponent gets none is still a possibility if we evaluate results after both tosses are complete. – verkter Jan 22 '17 at 3:54

$\underline{Answer\; for\; you\; getting\; at\; least\; one\; more\; head}$

Consider that initially, you both toss only one coin.

You can win only if you have already won, or are equal and win with your second toss.

Denoting your results in caps, and opponents in lowercase for clarity,

P(you win) = P(Ht) + P(HhH) + P(TtH) = $\dfrac12 + \dfrac14 + \dfrac14 = \dfrac12$

Interestingly, if you toss (n+1) coins against n tossed by your opponent, P(You win) is still $\dfrac12$
After tossing $n$ coins each, let $p$ be the probability that you are ahead. By symmetry, $p$ is also the probability that your opponent is ahead, and the probability of a tie is $1-2p$. You have just two ways to win: either you are ahead before the last toss, or there is a tie and you then get $H$.
Thus P(You win) $= p + (1-2p)\cdot\frac 12 = p+\frac 12 -p =\frac 12$
• Will it affect the results if we label one of your coins 1 and one of them 2 ? I have added material that explains your win probability remains 1/2 even if you have $(n+1)$ coins against your opponent's $n$. I think it should make it very clear. – true blue anil Jan 22 '17 at 4:00


# Show that the basis for an inner product space $V$ is an orthonormal basis

Suppose that $$\{v_1,...,v_n\}$$ be a basis for an inner product space $$V$$, and $$v=a_1v_1+...+a_nv_n$$ implies $$||v||^2=a_1^2+...+a_n^2$$, then can we say that the given basis is orthonormal? If so how?

I tried that $$=\sum_{i,j}a_ia_j$$ given that it is equal to $$a_1^2+...+a_n^2$$, thus can we conclude?

Hint: Note that $$\left\| v\right\|^2 = \langle v , v\rangle$$ for all $$v\in V$$. Therefore, you can show using inner product properties that $$\left\| v + w\right\|^2 = \left\|v \right\|^2 + \left\|w \right\|^2 + 2 \langle v, w\rangle$$ in a real inner product space, so

$$\langle v, w\rangle = \frac{\left\| v + w\right\|^2 - \left\|v \right\|^2 - \left\|w \right\|^2}{2},$$

for all $$v,w\in V$$.

Try to use this to show that $$\langle v_i , v_i\rangle = 1$$ for all $$i$$ and $$\langle v_i , v_j\rangle = 0$$ for all $$i\ne j$$.

• I am not getting how to apply this concept – RIYASUDHEEN TK 9747408592 Jun 8 at 5:25
• OK. Another hint: Note that $v_i + v_i = 2v_i$. Hence $\left\| v_i + v_i \right\|^2 = 2^2 = 4$ (using the assumption in the question). Similarly, find $\left\| v_i\right\|^2$ using the question's assumption, and then you have everything you need to find $\langle v_i, v_i \rangle$ using the equation in my answer. Then do a similar thing to find $\langle v_i, v_j\rangle$. – Minus One-Twelfth Jun 8 at 5:27

The hypothesis implies $$\|v_i\|^{2}=1$$ for all $$i$$ so each $$v_i$$ has norm $$1$$. Now consider $$\|v_i+v_j\|^{2}$$ where $$i \neq j$$. We get $$\|v_i+v_j\|^{2}=1^{2}+1^{2}=2$$. Expanding LHS in terms of the inner product we get $$\|v_i\|^{2}+\|v_i\|^{2}+2\langle v_i,v_j \rangle =2$$ which gives $$\langle v_i,v_j \rangle=0$$. This is the proof when teh scalare are real . I will leave the complex case to you.

• Ok i got it, thnk u sir – RIYASUDHEEN TK 9747408592 Jun 8 at 5:34
• In that case you can consider approving the answer by clicking on the tick mark. – Kabo Murphy Jun 8 at 5:46


# 17.5: Degree Distribution

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

Another local topological property that can be measured locally is, as we discussed already, the degree of a node. But if we collect them all for the whole network and represent them as a distribution, it will give us another important piece of information about how the network is structured:

A degree distribution of a network is a probability distribution

$P(k) =\frac{| \begin{Bmatrix} i | deg(i) =k \end{Bmatrix}| }{ n}. \label{(17.28)}$

i.e., the probability for a node to have degree $$k$$.

The degree distribution of a network can be obtained and visualized as follows:

The result is shown in Fig. 17.5.1.

Figure $$\PageIndex{1}$$: Visual output of Code 17.13.

You can also obtain the actual degree distribution P(k) as follows:

This list contains the value of (unnormalized) $$P(k) for k = 0,1,...,k_{max}$$, in this order. For larger networks, it is often more useful to plot a normalized degree histogram list in a log-log scale:

The result is shown in Fig. 17.5.2, which clearly illustrates differences between the three network models used in this example. The Erd˝os-R´enyi random network model has a bell-curved degree distribution, which appears as a skewed mountain in the log-log scale (blue solid line). The Watts-Strogatz model is nearly regular, and thus it has a very sharp peak at the average degree (green dashed line; $$k = 10\0 in this case). The Barab´asi-Albert model has a power-law degree distribution, which looks like a straight line with a negative slope in the log-log scale (red dotted line). Moreover, it is often visually more meaningful to plot not the degree distribution itself but its complementary cumulative distribution function (CCDF), deﬁned as follows: $F(k) =\sum_{k'=k} ^{\infty} P(k') \label{(17.29)}$ This is a probability for a node to have a degree \(k$$ or higher. By deﬁnition, $$F(0) = 1$$ and $$F(k_{max} + 1) = 0$$, and the function decreases monotonically along $$k$$. We can revise Code 17.15 to draw CCDFs:

Figure $$\PageIndex{2}$$: Visual output of Code 17.15.

In this code, we generate ccdf’s from Pk by calculating the sum of Pk after dropping its ﬁrst k entries. The result is shown in Fig. 17.5.2.
As you can see in the ﬁgure, the power law degree distribution remains as a straight line in the CCDF plot too, because $$F(k)$$ will still be a power function of$$k$$, as shown below:

Figure $$\PageIndex{3}$$: Visual output of Code 17.16.

F(k ) \begin{align} \sum_{k'=k} ^{\infty} {P(k')} = \sum_{k'=k} ^{infty}{ak' ^{-\gamma}} \label{(17.30)} \\ \approx \int_{k}^{infty} ak^{-\gamma}dk' = {\begin{bmatrix} \frac{ak^{J-\gamma+1}}{-\gamma + 1}\end{bmatrix}}^{\infty}_{k} = \frac{0-ak^{-\gamma+1}}{-\gamma +1} \label{(17.31)} \\ = \frac{a}{-\gamma -1}k ^{-(\gamma -1)} \label{(17.32)} \end{align}
This result shows that the scaling exponent of $$F(k)$$ for a power law degree distribution is less than that of the original distribution by 1, which can be visually seen by comparing their slopes between Figs. 17.5.2 and 17.5.3.

Exercise $$\PageIndex{1}$$

Import a large network data set of your choice from Mark Newman’s Network Data website: http://www-personal.umich.edu/~mejn/netdata/

Plot the degree distribution of the network, as well as its CCDF. Determine whether the network is more similar to a random, a small-world, or a scale-free network model.

If the network’s degree distribution shows a power law behavior, you can estimate its scaling exponent from the distribution by simple linear regression. You should use a CCDF of the degree distribution for this purpose, because CCDFs are less noisy than the original degree distributions. Here is an example of scaling exponent estimation applied to a Barab´asi-Albert network, where the linregress function in SciPy’s stats module is used for linear regression:

In the second code block, the domain and ccdf were converted to log scales for linear ﬁtting. Also, note that the original ccdf contained values for all k’s, even for those for which $$P(k) = 0$$. This would cause unnecessary biases in the linear regression toward the higher k end where actual samples were very sparse. To avoid this, only the data points where the value of $$F$$ changed (i.e., where there were actual nodes with degree k) are collected in the logkdata and logFdata lists.

The result is shown in Fig. 17.5.4, and also the following output comes out to the terminal, which indicates that this was a pretty good ﬁt:

Figure $$\PageIndex{4}$$: Visual output of Code 17.17.
According to this result, the CCDF had a negative exponent of about -1.97. Since this value corresponds to$$−(γ−$$1$$)$$, the actual scaling exponent $$γ$$ is about 2.97, which is pretty close to its theoretical value, 3.

Exercise $$\PageIndex{2}$$

Obtain a large network data set whose degree distribution appears to follow a power law, from any source (there are tons available online, including Mark Newman’s that was introduced before). Then estimate its scaling exponent using linear regression.


# Proof that $6^n$ always has a last digit of $6$

Without being proficient in math at all, I have figured out, by looking at series of numbers, that $6$ in the $n$-th power always seems to end with the digit $6$.

Anyone here willing to link me to a proof?

I've been searching google, without luck, probably because I used the wrong keywords.

-
@Stijn: your comment seems rather obnoxious. The OP did not say that he has "proved" anything; he made an observation which he thinks "seems to always" hold. And he's right... –  Pete L. Clark Sep 6 '11 at 0:47
@Stijn: Ragnar said "seems", so s/he knows it's not a proof. :) –  Guess who it is. Sep 6 '11 at 1:19
@J.M. A p.c. "s/he" in combination with a person called Ragnar is making me chuckle... –  t.b. Sep 6 '11 at 1:27
@Theo: I've been "victimized" by ladies using "manly" names on the Internet, so I'm covering myself just in case. :D –  Guess who it is. Sep 6 '11 at 1:34
My comment wasn't meant to be obnoxious. I'll delete it if it comes across as such. –  Stijn Sep 6 '11 at 8:40

We can prove it using mathematical induction.

Claim: $6^n\equiv 6\bmod 10$ for all $n\in\mathbb{N}$ (the symbol $\mathbb{N}$ denotes the natural numbers, and $\bmod 10$ means we are using modular arithmetic with a modulus of 10).

Base case (i.e., showing it's true for $n=1$): $$6^1\equiv 6\bmod 10\qquad\checkmark$$

Induction step (i.e., showing that, if it is true for $n=k$, then it is true for $n=k+1$):

$$6^k\equiv 6\bmod 10\implies 6^{k+1}\equiv 6^k\cdot 6\equiv6\cdot 6\equiv 36\equiv 6\bmod 10\qquad\qquad\checkmark$$

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If you multiply any two integers whose last digit is 6, you get an integer whose last digit is 6: $$\begin{array} {} & {} & {} & \bullet & \bullet & \bullet & \bullet & \bullet & 6 \\ \times & {} & {} &\bullet & \bullet & \bullet & \bullet & \bullet & 6 \\ \hline {} & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & 6 \end{array}$$ (Get 36, and carry the "3", etc.)

To put it another way, if the last digit is 6, then the number is $(10\times\text{something}) + 6$. So \begin{align} & {} \qquad \Big((10\times\text{something}) + 6\Big) \times \Big((10\times\text{something}) + 6\Big) \\ & = \Big((10\times\text{something})\times (10\times\text{something})\Big) \\ & {} \quad + \Big((10\times\text{something})\times 6\Big) + \Big((10\times\text{something})\times 6\Big) + 36 \\ & = \Big(10\times \text{something}\Big) +36 \\ & = \Big(10\times \text{something} \Big) + 6. \end{align}

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HINT $\rm\ \ 6-1\ |\ 6^k-1,\$ so $\rm\:\ 2,5\ |\ 6^n-6\ \Rightarrow\ 10\ |\ 6^n - 6\:,\$ i.e. $\rm\ 6^n\ =\ 6 + 10\ k\:$ for $\rm\:k\in\mathbb Z\:.$

Alternatively: $\rm\ mod\ 10:\ \ 6^n\equiv 6\$ since it is $\rm\ 0^n \equiv 0\pmod 2,\ \ 1^n \equiv 1\pmod 5$

Similarly odd $\rm\:b\: \Rightarrow\: (b+1)^n\equiv b+1\pmod{2\:b}\:,\:$ so $\rm\:(b+1)^n\:$ has last digit $\rm\:b+1\:$ in radix $\rm\:2\:b\:.$

NOTE how modular arithmetic reduces the induction to the trivial inductions $\rm\ 0^n = 0,\ 1^n = 1\:.$ This is a prototypical example of the sort of simplification afforded by reducing arithmetical problems to their counterparts in the simpler arithmetical rings of integers $\rm\:(mod\ m)\:.\:$

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In some contexts, this would be a good way to answer this question. But given the way the question was phrased on this occasion, I wouldn't have considered it probable that this is one of those. –  Michael Hardy Sep 6 '11 at 1:34
@Mic Surely the OP can grok the first proof. The rest requires only basic knowledge of modular arithmetic - which it seems is known to the OP given the accepted answer. Even if was not known to the OP, it is known to many other readers. The site is for all to learn - not just OP's. So I disagree. –  Bill Dubuque Sep 6 '11 at 1:58
Thanks for the answer, I understand it. Actually, I stumbled upon seeing the series, trying to prove $5 | 6^k-1$. My professor has since pointed me in the right direction, using mathematical induction. –  Ragnar123 Sep 6 '11 at 18:20
@Ragnar Thanks for the feedback. Should anything I write be unclear, please feel free to ask further questions. I am always happy to elaborate. –  Bill Dubuque Sep 6 '11 at 18:35

This follows from the more general result that the product of two numbers ending with digit 6 also ends with digit 6. This can be proved in an elementary way: $$(10x+6)\cdot(10y+6) = 100xy + 60x +60y + 36 = 10(10xy+6x +6y +3) + 6 = 10z+6$$

Of course, avoiding all these letters is what congruences are all about.

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On the other hand, for the purposes of this question, I prefer this to the mod solutions. :) –  Guess who it is. Sep 6 '11 at 1:21
BTW, the same holds for numbers ending with 1 or 5, by the same reasoning. –  lhf Sep 6 '11 at 1:51

$6 \times 6 \equiv 6 \pmod{10}$.

Or, more elementarily put, think back to the pen-and-paper multiplication algorithm. When you multiply something by 6, the only part of the original number that can affect the last digit of the result is the last digit of the original. If you start with something that ends in 6, you get 36 for the last position, write 6 down and carry the 3. But no matter what happens after the carry, it cannot affect the final 6 that you've already produced.

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