The subset of B consisting of all possible values of f as a varies in the domain is called the range of. 1 T(~x + ~y) = T(~x) + T(~y)(preservation of addition) 2 T(a~x) = aT(~x)(preservation of scalar multiplication) Linear Transformations: Matrix of a Linear Transformation Linear Transformations Page 2/13. A basis for the kernel of L is {1} so the kernel has dimension 1. Matrix vector products as linear transformations. All Slader step-by-step solutions are FREE. Inversion: R(z) = 1 z. Let A = 2 4 0. Find the kernel of the linear transformation L: V→W. Proof: This theorem is a proved in a manner similar to how we solved the above example. range(T)={A in W | there exists B in V such that T(B)=A}. The $$\textit{nullity}$$ of a linear transformation is the dimension of the kernel, written $$nul L=\dim \ker L. Thus, we should be able to find the standard matrix for. It is essentially the same thing here that we are talking about. Let T: V !W be a linear transformation. It relates the dimension of the kernel and range of a linear map. In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping V → W between two modules (for example, two vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication. Construct a linear transformation f and vector Y so that the system takes the form f(X)=Y. I tried to RREF it on my calculator but it says invalid dim, I am not sure of what to do and all the examples I have looked up are for square matrices, any help would be appreciated, thanks!. And we saw that earlier in the video. Suppose T : V !W is a linear transformation. In Section 4, we deﬁne the kernel whitening transformation and orthogonalize non-. Before we look at some examples of the null spaces of linear transformations, we will first establish that the null space can never be equal to the empty set, in. The confidence of the interval [107, 230] is less than 95%. In both cases, the kernel is the set of solutions of the corresponding homogeneous linear equations, AX = 0 or BX = 0. The function of kernel is to take data as input and transform it into the required form. 0004 From the previous lesson, we left it off defining what the range of a linear map is. be a linear transformation. Specifically, if T: n m is a linear transformation, then there is a unique m n matrix, A, such that T x Ax for all x n. Griti is a learning community for students by students. Algebra Linear Algebra: A Modern Introduction 4th Edition In Exercises 5-8, find bases for the kernel and range of the linear transformations T in the indicated exercises. " • The fact that T is linear is essential to the kernel and range being subspaces. LTR-0060: Isomorphic Vector Spaces We define isomorphic vector spaces, discuss isomorphisms and their properties, and prove that any vector space of dimension is isomorphic to. (a) Find a basis of the range of P. the kernel of a a linear transformation is the set of vectors in the null space of the matrix for that linear transformation. Kernel, image, nullity, and rank Math 130 Linear Algebra D Joyce, Fall 2015 De nition 1. Linear Transformation. Since the nullity is the dimension of the null space, we see that the nullity of T is 0 since the dimension of the zero vector space is 0. The following examples illustrate the syntax. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. To prove the transformation is linear, the transformation must preserve scalar multiplication, addition, and the zero vector. We then consider invertible linear transformations, and then use the resulting ideas to prove the rather stunning result that (in a very precise sense). Vector Spaces and Linear Transformations Beifang Chen Fall 2006 1 Vector spaces A vector space is a nonempty set V, whose objects are called vectors, equipped with two operations, called addition and scalar multiplication: For any two vectors u, v in V and a scalar c, there are unique vectors u+v and cu in V such that the following properties are satisﬂed. Kernel trick allows us to transform the data in high dimensional (potentially inﬁnite) using inner products, without actually using the non linear feature mapping. SUBSCRIBE to the channel and. We discuss the kernel and range of linear transformations, and then prove that the range of a linear transformation is a subspace. What is the outcome of solving the problem?. The Gaussian is a self-similar function. Sums and scalar multiples of linear transformations. This MATLAB function returns predicted class labels for each observation in the predictor data X based on the binary Gaussian kernel classification model Mdl. Find the kernel of f. Then the Kernel of the linear transformation T is all elements of the vector space V that get mapped onto the zero element of the vector space W. In Section 3, we compute the whitening transformation. We show that for high-dimensional data, a particular framework for learning a linear transformation of the data based on the LogDet divergence can be efficiently kernelized to learn a metric (or equivalently, a kernel function) over an. T is a linear transformation. The problem comes when finding the Kernel basis. Define the linear transformation T(x) = A * x for A an m by n matrix. T(x 1,x 2,x 3,x 4)=(x 1−x 2+x 3+x 4,x 1+2x 3−x 4,x 1+x 2+3x 3. We denote the kernel of T by ker(T) or ker(A). Now, we connect together all the ideas we’ve talked. To find the null space we must first reduce the #3xx3# matrix found above to row echelon form. Linear transformations as matrix vector products. S: ℝ3 → ℝ3. Define pre-image of U, denoted T -1. Griti is a learning community for students by students. One-to-One linear transformations: In college algebra, we could perform a horizontal line test to determine if a function was one-to-one, i. De nition 1. + for all vectors VI, for all scalars Cl, F(cv) for all scalars c, for all ve V, for all A function F: V —W is linear W be a subspace of Rk Let V be a subspace of Let it respects the linear operations,. linear transformation. Up Main page Definition. In particular, there exists a nonzero solution. Let V;W be vector spaces over a eld F. 2 The kernel and range of a linear transformation. Most off-the-shelf classifiers allow the user to specify one of three popular kernels: the polynomial, radial basis function, and sigmoid kernel. The same considerations apply to rows as well as columns. For example linear, nonlinear, polynomial, radial basis function. Recall that if a set of vectors v 1;v 2;:::;v n is linearly independent, that means that the linear combination c. The null space of T, denoted N(T), is de ned as N(T) := fv2V: T(v) = 0g: Remark 3. Let T: R n → R m be a linear transformation. Find the kernel of the linear transformation. If T : Rm → Rn is a linear transformation, then the set {x | T(x) = 0 } is called the kernelof T. To see why image relates to a linear transformation and a matrix, see the article on linear. Note that the squares of s add, not the s 's themselves. 2 Kernel and Range of a Linear Transformation Performance Criteria: 2. IV Image and Kernel of Linear Transformations Motivation: In the last class, we looked at the linearity of this function: F : R3 R2 F(x,y,z)=(x+y+z,2x-3y+4z) How does a 3 dimensional space get ‘mapped into’ a 2 dimensional space? At least one dimension ‘collapses’, or disappears. Let T: V !Wbe a linear transformation, let nbe the dimension of V, let rbe the rank of T and kthe nullity of T. Because Tis one-to-one, the dimension of the image of Tmust be n. Gaussian Kernel always provides a value between 0 and 1. Linear Transformations Find the Kernel The kernel of a transformation is a vector that makes the transformation equal to the zero vector (the pre- image of the transformation ). Corollary 2. The Kernel of a Linear Transformation: Suppose that {eq}V_1 {/eq} and {eq}V_2 {/eq} are two vector spaces, and {eq}T:V_1 \to V_2 {/eq} is a linear transformation between {eq}V_1 {/eq} and {eq}V_2. For instance, for m = n = 2, let A = • 1 2 1 3 ‚; B = • 2 1 2 3 ‚; X = • x1 x2 x3 x4 ‚: Then F: M(2;2)! M(2;2) is given by F(X) = • 1 2 1 3 ‚• x1 x2 x3 x4 ‚• 2 1 2 3 ‚ = • 2x1 +2x2 +4x3 +4x4 x1 +3x2 +2x3 +6x4 2x1 +2x2 +6x3 +6x4 x1 +3x2 +3x3 +9x4 ‚: (b) The function D: P3! P2, deﬂned by D ¡ a0 +a1t+a2t 2 +a 3t 3 ¢ = a1 +2a2t+3a3t2; is a linear transformation. Notation: f: A 7!B If the value b 2 B is assigned to value a 2 A, then write f(a) = b, b is called the image of a under f. The dimension of the kernel of T is the same as the dimension of its null space and is called the nullity of the transformation. Support vector machine with a polynomial kernel can generate a non-linear decision boundary using those polynomial features. The offset c determines the x-coordinate of the point that all the lines in the posterior go though. Find the kernel of the linear transformation L: V→W. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. The transformation defines a map from ℝ3 to ℝ3. (b) Find the matrix representation of L with respect to the standard basis 1;x;x2. Convolution with a Gaussian is a linear operation, so a convolution with a Gaussian kernel followed by a convolution with again a Gaussian kernel is equivalent to convolution with the broader kernel. First here is a definition of what is meant by the image and kernel of a linear transformation. Suppose L∶V → W is a linear isomorphism then it is a bijection. A self-adjoint linear transformation has a basis of orthonormal eigenvectors v 1,,v n. The kernel of 𝐴 is calculated by finding the reduced echelon form of this matrix using Gauss-Jordan elimination and then writing the solution in a particular way. RHS of equation is a 2 row by 1 column matrix. {\mathbb R}^n. Use the kernel and image to determine if a linear transformation is one to one or onto. (The dimension of the image space is sometimes called the rank of T, and the dimension of the kernel is sometimes called the nullity of T. Finding matrices such that M N = N M is an important problem in mathematics. Non Linear SVM using Kernel. Kernel algorithms using a linear kernel are often equivalent to their non-kernel counterparts, i. The kernel or null-space of a linear transformation is the set of all the vectors of the input space that are mapped under the linear transformation to the null vector of the output space. SVM algorithms use a set of mathematical functions that are defined as the kernel. T(v) = Av represents the linear transformation T. Let T: V !Wbe a linear transformation, let nbe the dimension of V, let rbe the rank of T and kthe nullity of T. Find a basis of the null space of the given m x n matrix A. De ne T : P 2!R2 by T(p) = p(0) p(0). 3, -3 , 1] Find the basis of the image of a linear transformation T defined by T(x)=Ax. the kernel of a a linear transformation is the set of vectors in the null space of the matrix for that linear transformation. Corollary 2. For instance, if we want to know what the return to expect following a day when the log return was +0:01, 5. A= 0 1 −1 0. Suppose T:R^3 \\to R^3,\\quad T(x,y,z) = (x + 2y, y + 2z, z + 2x) Part of Solution: The problem is solved like this: A =. ) T: R^2 rightarrow R^2, T(x, y) = (x + 2y, y - x) ker(T) = {: x, y R} T(v) = Av represents the linear transformation T. Because is a composition of linear transformations, itself is linear (Theorem th:complinear of LTR-0030). So,wehave w 1 = v1 kv1k = 1 √ 12 +12. The event times that satisfy include 107, 109, 110, 122, 129, 172, 192, 194, and 230. Find the matrix of the orthogonal projection onto W. Let T be a linear transformation from Rm to Rn with n × m matrix A. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. It is essentially the same thing here that we are talking about. Sources of subspaces: kernels and ranges of linear transformations. be a linear transformation. Non Linear SVM using Kernel. TRUE Remember that Ax gives a linear combination of columns of A using x entries as weights. The null space of T, denoted N(T), is de ned as N(T) := fv2V: T(v) = 0g: Remark 3. Affine transformations", you can find examples of the use of linear transformations, which can be defined as a mapping between two vector spaces that preserves linearity. Find the kernel of the linear transformation. The following is a basic list of model types or relevant characteristics. SUBSCRIBE to the channel and. 2 (The Kernel and Range)/3. These are all vectors which are annihilated by the transformation. Then rangeT is a ﬁnite-dimensional subspace of W and dimV = dimnullT +dimrangeT. (f) The set of all solutions of a homogeneous linear differential equation is the kernel of a linear transformation. Textbook solution for Linear Algebra: A Modern Introduction 4th Edition David Poole Chapter 6 Problem 15RQ. 2-T:R 3 →R 3,T(x,y,z)=(x,0,z). Introduction to Linear Algebra exam problems and solutions at the Ohio State University. 4 LECTURE 7: LINEAR TRANSFORMATION We have L(v) = 0W = L(0V). 2 Kernel and Range of linear Transfor-mation We will essentially, skip this section. Find the matrix of the given linear transformation T with respect to the given basis. Here we consider the case where the linear map is not necessarily an isomorphism. Before we do that, let us give a few deﬁnitions. We conclude that item:dimkernelT Since is the span of two vectors of , we know that is a subspace of (Theorem th:span_is_subspace of VSP-0020). It doesn't hurt to have it, but it isn't necessary here (in finding the kernel). You can even pass in a custom kernel. Neal, WKU Theorem 2. For two linear transformations K and L taking Rn Rn , and v Rn , then in general K(L(v)) = L(K(v)). Section 2 describes the calculation of the canonical angles. In this section, you will learn most commonly used non-linear regression and how to transform them into linear regression. If there are page buffers, the total number of bytes in the page buffer area is 'data_len'. Find the matrix of the given linear transformation T with respect to the given basis. And we saw that earlier in the video. Find the kernel of the linear transformation. , the solutions of the equation A~x = ~ 0. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. In the case where V is finite-dimensional, this implies the rank-nullity theorem:. Remarks I The kernel of a linear transformation is a. (The dimension of the image space is sometimes called the rank of T, and the dimension of the kernel is sometimes called the nullity of T. The range of A is the columns space of A. The linear transformation t 1 is the orthogonal reflection in the line y = x. Then (1) is a subspace of. SPECIFY THE VECTOR SPACES Please select the appropriate values from the popup menus, then click on the "Submit" button. The challenge is to find a transformation -> , such that the transformed dataset is linearly separable in. Recall: Linear Transformations De nition A transformation T : Rn!Rm is alinear transformationif it satis es the following two properties for all ~x;~y 2Rn and all (scalars) a 2R. 3 (Nullity). We have some fundamental concepts underlying linear transformations, such as the kernel and the image of a linear transformation, which are analogous to the zeros and range of a function. the kernel of a a linear transformation is the set of vectors in the null space of the matrix for that linear transformation. 4 Linear Transformations The operations \+" and \" provide a linear structure on vector space V. Analysis & Implementation Details. SPECIFY THE VECTOR SPACES Please select the appropriate values from the popup menus, then click on the "Submit" button. Conversely any linear fractional transformation is a composition of simple trans-formations. But, if we apply transformation X² to get: New Feature: X = np. If T(~x) = A~x, then the kernel of T is also called the kernel of A. Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Sx— 15y +4z x. Support vector machine with a polynomial kernel can generate a non-linear decision boundary using those polynomial features. The disadvantages are: 1) If the data is linearly separable in the expanded feature space, the linear SVM maximizes the margin better and can lead to a sparser solution. 1 LINEAR TRANSFORMATIONS 217 so that T is a linear transformation. Then, which of the following statements is always true?. The nullspace of a linear operator A is N(A) = {x ∈ X: Ax = 0}. Preimage and kernel example. The range of L is the set of all vectors b ∈ W such that the equation L(x) = b has a solution. Summary: Kernel 1. Define the transformation \Omega: L(V,W) \to M_{m \times n} (\mathbb{R}) Stack Exchange Network Stack Exchange network consists of 176 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Gaussian Kernel always provides a value between 0 and 1. We write ker(A) or ker(T). However, there is also a limited amount of support for working with sparse matrices and vectors. im (T): Image of a transformation. on the order of 1000 or less since the algorithm is cubic in the number of features. And if the transformation is equal to some matrix times some vector, and we know that any linear transformation can be written as a matrix vector product, then the kernel of T is the same thing as the null space of A. Note that the range of the linear transformation T is the same as the range of the matrix A.$$ Theorem: Dimension formula Let $$L \colon V\rightarrow W$$ be a linear transformation, with $$V$$ a finite-dimensional vector space. N(T) is also referred to as the kernel of T. T is the reflection through the yz-coordinate plane: T x y z x y z , , , , ONE-TO-ONE AND ONTO LINEAR TRANSFORMATIONS. Let T: R 3!R3 be the transformation on R which re ects every vector across the plane x+y+z= 0. Find the kernel of the linear transformation. How to find the kernel of a linear transformation? Let B∈V =Mn(K) and let CB :V →V be the map defined by CB(A)=AB−BA. Then the kernel of L is de ned to be: ker(L) := fv 2V : L(v) = ~0g V i. The kernel of A are all solutions to the linear system Ax = 0. Question: Why is a linear transformation called “linear”?. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. This MATLAB function returns predicted class labels for each observation in the predictor data X based on the binary Gaussian kernel classification model Mdl. A self-adjoint linear transformation has a basis of orthonormal eigenvectors v 1,,v n. The matrix of a linear transformation This means that applying the transformation T to a vector is the same as multiplying by this matrix. It doesn't hurt to have it, but it isn't necessary here (in finding the kernel). This makes it possible to "turn around" all the arrows to create the inverse linear transformation $\ltinverse{T}$. Thus, the kernel consists of all matrices of the form [a b] [0 a] for a, b ∈ K; hence the nullity = 2. nan_euclidean_distances (X) Calculate the euclidean distances in the presence of missing values. Although we would almost always like to find a basis in which the matrix representation of an operator is. KPCA with linear kernel is the same as standard PCA. Then the Kernel of the linear transformation T is all elements of the vector space V that get mapped onto the zero element of the vector space W. If T isn't an isomorphism find bases of the kernel and image of T, and. Transformation Matrices. Now, let $\phi: V\longrightarrow W$ be a linear mapping/transformation between the two vector spaces. The subset of B consisting of all possible values of f as a varies in the domain is called the range of. Construct a linear transformation f and vector Y so that the system takes the form f(X)=Y. The next theorem is the key result of this chapter. The following examples illustrate the syntax. Two examples of linear transformations T :R2 → R2 are rotations around the origin and reﬂections along a line through the origin. Definition of the Image of linear map 𝐋. large values of , and clearly approach the linear regression; the curves shown in red are for smaller values of. Polynomial Kernel. Note that the range of the linear transformation T is the same as the range of the matrix A. Find a basis for the kernel of T and the range of T. Let’s begin by rst nding the image and kernel of a linear transformation. This paper studies the conditions for the idempotent transformation and the idempotent rank transformation direct sum decomposition for finite dimension of linear space. The following examples illustrate the syntax. Now is the time to redefine your true self using Slader’s free Linear Algebra: A Modern Introduction answers. Definition Kernel and Image. These functions can be different types. To connect linear algebra to other fields both within and without mathematics. KPCA with linear kernel is the same as standard PCA. This mapping is called the orthogonal projection of V onto W. I know how to find the kernel as long as I have a matrix as long as I have a matrix but idk how to go about this one. (d)The rank of a linear transformation equals the dimension of its kernel. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. (b)Find a linear transformation whose kernel is Sand whose range is S?. Given two vector spaces V and W and a linear transformation L : V !W we de ne a set: Ker(L) = f~v 2V jL(~v) = ~0g= L 1(f~0g) which we call the kernel of L. There are some important concepts students must master to solve linear transformation problems, such as kernel, image, nullity, and rank of a linear transformation. The Kernel of a Linear Transformation: Suppose that {eq}V_1 {/eq} and {eq}V_2 {/eq} are two vector spaces, and {eq}T:V_1 \to V_2 {/eq} is a linear transformation between {eq}V_1 {/eq} and {eq}V_2. In this paper, we study metric learning as a problem of learning a linear transformation of the input data. Problem: I can't find answer to a problem. (e)The nullity of a linear transformation equals the dimension of its range. Since the nullity is the dimension of the null space, we see that the nullity of T is 0 since the dimension of the zero vector space is 0. 17 The rank of a linear map is less than or equal to the dimension of the domain. IV Image and Kernel of Linear Transformations Motivation: In the last class, we looked at the linearity of this function: F : R3 R2 F(x,y,z)=(x+y+z,2x-3y+4z) How does a 3 dimensional space get ‘mapped into’ a 2 dimensional space? At least one dimension ‘collapses’, or disappears. 2 Kernel and Range of linear Transfor-mation We will essentially, skip this section. (a) Find a basis of the range of P. Describe the kernel and range of a linear transformation. The image of a linear transformation or matrix is the span of the vectors of the linear transformation. Morphological transformations are some simple operations based on the image shape. (c) Find the nullity and rank of P. sage : M = MatrixSpace ( IntegerRing (), 4 , 2 )( range ( 8 )) sage : M. Then, ker(L) is a subspace of V. We will start with Hinge Loss and see how the optimization/cost function can be changed to use the Kernel Function,. If M is singular there must be a linear combination of rows of M that sums to the zero row vector. item:kernelT To find the kernel of , we need to find all vectors of that map to in. 2 The kernel and range of a linear transformation. The set consisting of all the vectors v 2V such that T(v) = 0 is called the kernel of T. This can be defined set-theoretically as follows:. MATH 316U (003) - 10. S: ℝ3 → ℝ3. Therefore, w 1 and w 2 form an orthonormal basis of the kernel of A. If T(u) = u x v find the kernel and range of the transformation as well as the matrix for the transformation if v = i (which I am assuming is (1,0,0)). to construct the whitening transformation matrix for orthogonalizing the linear subspaces in the feature space F. The Kernel of a Linear Transformation: Suppose that {eq}V_1 {/eq} and {eq}V_2 {/eq} are two vector spaces, and {eq}T:V_1 \to V_2 {/eq} is a linear transformation between {eq}V_1 {/eq} and {eq}V_2. Before we do that, let us give a few deﬁnitions. 0000 Today we are going to continue our discussion of the kernel and range of a linear map of a linear transformation. Finding the kernel of a linear transformation involving an integral. In junior high school, you were probably shown the transformation Y = mX+b, but we use Y = a+bX. range(T)={A in W | there exists B in V such that T(B)=A}. Algebra Examples. {\mathbb R}^m. Other Kernel Methods •A lesson learned in SVM: a linear algorithm in the feature space is equivalent to a non-linear algorithm in the input space •Classic linear algorithms can be generalized to its non-linear version by going to the feature space –Kernel principal component analysis, kernel independent component analysis, kernel. Deﬁnition 6. Some linear transformations possess one, or both, of two key properties, which go by the names injective and surjective. transformation, the kernel and the image. Find more Mathematics widgets in Wolfram|Alpha. Note: It is convention to use the Greek letter 'phi' for this transformation , so I'll use. Lesson: Image and Kernel of Linear Transformation Mathematics. Determine whether T is an isomorphism. ∆ Let T: V ‘ W be a linear transformation, and let {eá} be a basis for V. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. We illustrated the quadratic kernel in quad-kernel. , the solutions of the equation A~x = ~ 0. Linear algebra - Practice problems for midterm 2 1. We have some fundamental concepts underlying linear transformations, such as the kernel and the image of a linear transformation, which are analogous to the zeros and range of a function. This paper studies the conditions for the idempotent transformation and the idempotent rank transformation direct sum decomposition for finite dimension of linear space. More importantly, as an injective linear transformation, the kernel is trivial (Theorem KILT), so each pre-image is a single vector. To take an easy example, suppose we have a linear transformation on R 2 that maps (x, y) to (4x+ 2y, 2x+ y). KERNEL AND RANGE OF LINEAR TRANSFORMATION199 6. The transformation is selected from a parametric family, which is allowed to be quite general in our theoretical study. For a linear transformation T from Rn to Rm, † im(T) is a subset of the codomain Rm of T, and † ker(T) is a subset of the domain Rn. Finding a basis of the null space of a matrix. Find the matrix of the given linear transformation T with respect to the given basis. Note that N(T) is a subspace of V, so its dimension can be de ned. We solve by finding the corresponding 2 x 3 matrix A, and find its null space and column span. Such a repre-sentation is frequently called a canonical form. Let V;W be vector spaces over a eld F. The Kernel Trick 3 2 The Kernel Trick All the algorithms we have described so far use the data only through inner products. manhattan_distances (X[, Y, …]) Compute the L1 distances between the vectors in X and Y. 0)( =vT ker( ) {v | (v) 0, v }T T V= = ∀ ∈. The kernel and image of a matrix A of T is defined as the kernel and image of T. If V is finite-dimensional, then so are Im(T) and ker(T), anddim(Im(T))+dim(ker(T))=dimV. Since the correlation coefficient is maximized when a scatter diagram is linear, we can use the same approach above to find the most normal transformation. Find the kernel of the linear transformation L: V→W. Determine whether the following functions are linear transformations. All Slader step-by-step solutions are FREE. IV Image and Kernel of Linear Transformations Motivation: In the last class, we looked at the linearity of this function: F : R3 R2 F(x,y,z)=(x+y+z,2x-3y+4z) How does a 3 dimensional space get ‘mapped into’ a 2 dimensional space? At least one dimension ‘collapses’, or disappears. However, there is also a limited amount of support for working with sparse matrices and vectors. We will start with Hinge Loss and see how the optimization/cost function can be changed to use the Kernel Function,. Then T is a linear transformation. {\mathbb R}^n. Discuss this quiz (Key: correct, incorrect, partially correct. Let’s begin by rst nding the image and kernel of a linear transformation. Find the matrix of the given linear transformation T with respect to the given basis. We prove the theorems relating to kernel and image of linear transformation. To connect linear algebra to other fields both within and without mathematics. The general solution is a linear combination of the elements of a basis for the kernel, with the coefficients being arbitrary constants. Gauss-Jordan elimination yields: Thus, the kernel of consists of all elements of the form:. Null space. A= [-3, -2 , 4. What is the outcome of solving the problem?. 1 Example Clearly, the data on the left in ﬁgure 1 is not linearly separable. If T(~x) = A~x, then the kernel of T is also called the kernel of A. How Linear Transformations Affect the Mean and Variance. Now is the time to redefine your true self using Slader’s free Linear Algebra: A Modern Introduction answers. Note that the squares of s add, not the s 's themselves. [Solution] To get an orthonormal basis of W, we use Gram-Schmidt process for v1 and v2. Step-by-Step Examples. The next example illustrates how to find this matrix. De nition 1. Finding the kernel of the linear transformation. Then the Kernel of the linear transformation T is all elements of the vector space V that get mapped onto the zero element of the vector space W. The only solution is x = y = 0, and thus the zero vector (0. (2) is injective if and only if. Of course we can. transformation, the kernel and the image. It is normally performed on binary images. Preimage and kernel example. The kernel of a linear operator is the set of solutions to T(u) = 0, and the range is all vectors in W which can be expressed as T(u) for some u 2V. SVG image not dispayed. Hello and welcome back to Educator. If a linear transformation T: R n → R m has an inverse function, then m = n. To find the null space we must first reduce the #3xx3# matrix found above to row echelon form. Let us say I have 3 vectors in v that map to 0, those three vectors, that is my kernel of my linear map. The null space of T, denoted N(T), is de ned as N(T) := fv2V: T(v) = 0g: Remark 3. Find the matrix of the orthogonal projection onto W. (b) Find the matrix representation of L with respect to the standard basis 1;x;x2. 1 Matrix Linear Transformations Every m nmatrix Aover Fde nes linear transformationT A: Fn!Fmvia matrix multiplication. Create a system of equations from the vector equation. [Solution] To get an orthonormal basis of W, we use Gram-Schmidt process for v1 and v2. The kernel of L is the solution set of the homogeneous linear equation L(x) = 0. Basically, the kernel of a linear map, from a vector space v to a vector space w is all those vectors in v that map to the 0 vector. Find the kernel of the linear transformation L: V→W. visualize what the particular transformation is doing. The range of L is the set of all vectors b ∈ W such that the equation L(x) = b has a solution. , it can be applied to unseen data. The linear transformation t 2 is the orthogonal projection on the x-axis. 2 The kernel and range of a linear transformation. TRUE To show this we show it is a subspace Col A is the set of a vectors that can be written as Ax for some x. (c)The range of a linear transformation is a subspace of the co-domain. SUBSCRIBE to the channel and. To find the kernel, you just need to determine the dimensionality of the solution space to the linear system. The Gaussian is a self-similar function. This mapping is called the orthogonal projection of V onto W. The spline can also be used for prediction. The following charts show some of the ideas of non-linear transformation. Now, let $\phi: V\longrightarrow W$ be a linear mapping/transformation between the two vector spaces. Justify your answers. The disadvantages are: 1) If the data is linearly separable in the expanded feature space, the linear SVM maximizes the margin better and can lead to a sparser solution. To compute the kernel, find the null space of the matrix of the linear transformation, which is the same to find. Please wait until "Ready!" is written in the 1,1 entry of the spreadsheet. Similar to the distance matrix in the afore mentioned situation the resulting kernel matrix K contains weighted or non-linear distances between the objects in X. Then for any x ∞ V we have x = Íxáeá, and hence T(x) = T(Íxáeá) = ÍxáT(eá). It is the set of vectors, the collection of vectors that end up under the transformation mapping to 0. Find the rank and nullity of a linear transformation from R^3 to R^2. {\mathbb R}^m. There are some important concepts students must master to solve linear transformation problems, such as kernel, image, nullity, and rank of a linear transformation. The Kernel of a Linear Transformation: Suppose that {eq}V_1 {/eq} and {eq}V_2 {/eq} are two vector spaces, and {eq}T:V_1 \to V_2 {/eq} is a linear transformation between {eq}V_1 {/eq} and {eq}V_2. If there are page buffers, the total number of bytes in the page buffer area is 'data_len'. TThis quiz is designed to test your knowledge of linear transformations and related concepts such as rank, nullity, invertibility, null space, range, etc. SKBs are composed of a linear data buffer, and optionally a set of 1 or more page buffers. Define pre-image of U, denoted T -1. If T(~x) = A~x, then the kernel of T is also called the kernel of A. Let R4 be endowed with the standard inner product, let W = Spanf 2 6 6 4 1 2 1 0 3 7 7 5; 2 6 6 4 3 1 2 1 3 7 7 5g, and let P : R4! R4 be the orthogonal projection in R4 onto W. Find polynomial(s) p i(t) that span the kernel of T. The idea of a linear transformation is that one variable is mapped onto another in a 1-to-1 fashion. Theorem Let T:V→W be a linear transformation. {\mathbb R}^n. Course goals. Specifically, if T: n m is a linear transformation, then there is a unique m n matrix, A, such that T x Ax for all x n. We define the kernel of $\phi$ to be. Vector Spaces and Linear Transformations Beifang Chen Fall 2006 1 Vector spaces A vector space is a nonempty set V, whose objects are called vectors, equipped with two operations, called addition and scalar multiplication: For any two vectors u, v in V and a scalar c, there are unique vectors u+v and cu in V such that the following properties are satisﬂed. One thing to look out for are the tails of the distribution vs. – Suppose we have a linear transformation f: Rn!. For example linear, nonlinear, polynomial, radial basis function. {\mathbb R}^m. It is the set of vectors, the collection of vectors that end up under the transformation mapping to 0. What is the range of T in R2?. It doesn't hurt to have it, but it isn't necessary here (in finding the kernel). De nition 3. One-to-One linear transformations: In college algebra, we could perform a horizontal line test to determine if a function was one-to-one, i. The kernel of a transformation is a vector that makes the transformation equal to the zero vector (the pre-image of the transformation). Demonstrate: A mapping between two sets L: V !W. KERNEL AND RANGE OF LINEAR TRANSFORMATION199 6. Before we look at some examples of the null spaces of linear transformations, we will first establish that the null space can never be equal to the empty set, in. We describe the range by giving its basis. First here is a definition of what is meant by the image and kernel of a linear transformation. ) T: R^2 rightarrow R^2, T(x, y) = (x + 2y, y - x) ker(T) = {: x, y R} T(v) = Av represents the linear transformation T. Remarks I The kernel of a linear transformation is a. linear transformation. Step-by-Step Examples. TRUE Remember that Ax gives a linear combination of columns of A using x entries as weights. Note that the squares of s add, not the s 's themselves. Synonyms: kernel onto A linear transformation, T, is onto if its range is all of its codomain, not merely a subspace. More Examples of Linear Transformations: solutions: 6: More on Bases of $$\mathbb{R}^n$$, Matrix Products: solutions: 7: Matrix Inverses: solutions: 8: Coordinates: solutions: 9: Image and Kernel of a Linear Transformation, Introduction to Linear Independence: solutions: 10: Subspaces of $$\mathbb{R}^n$$, Bases and Linear Independence. Find the matrix of the given linear transformation T with respect to the given basis. There are some important concepts students must master to solve linear transformation problems, such as kernel, image, nullity, and rank of a linear transformation. Since Whas dimension n, the image of Tmust equal W. (c)Find a linear transformation whose kernel is S?and whose range is S. [Linear Algebra] Finding the kernel of a linear transformation. Similarly, we say a linear transformation T: max 1 i n di +2 r 2R2 n (p 2+ln r 1 )) 1 n+1 where the support of the distribution D is assumed to be contained in a ball of radius R. Explainer +9; Read. The following section goes through the the different objective functions and shows how to use Kernel Tricks for Non Linear SVM. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. It is an extension of Principal Component Analysis (PCA) - which is a linear dimensionality reduction technique - using kernel methods. You can even pass in a custom kernel. im (T): Image of a transformation. T [x, y, z, w] = [x + 2y + z - w] [2x + 3y - z + w] LHS of equation is a 4 row by 1 column matrix. It is given by the inner product plus an optional constant c. From this, it follows that the image of L is isomorphic to the quotient of V by the kernel:  ≅ /  (). (b) The dual space V ∗ of the vector space V is the set of all linear functionals on V. To find the kernel of a matrix A is the same as to solve the system AX = 0, and one usually does this by putting A in rref. Griti is a learning community for students by students. That is it. 3, -3 , 1] Find the basis of the image of a linear transformation T defined by T(x)=Ax. , Mladenov, M. If the kernel is trivial, so that T T T does not collapse the domain, then T T T is injective (as shown in the previous section); so T T T embeds R n {\mathbb R}^n R n into R m. We have step-by-step solutions for your textbooks written by Bartleby experts! Find the nullity of the linear transformation T : M n n → ℝ defined by T ( A ) = tr ( A ). De nition 1. A = [2 1] [3 4]. The kernel of a linear transformation is a vector space. Suppose T:R^3 \\to R^3,\\quad T(x,y,z) = (x + 2y, y + 2z, z + 2x) Part of Solution: The problem is solved like this: A =. Solving systems of nonlinear equa- tions can be tricky. ker(T)={A in V | T(A)=0} The range of T is the set of all vectors in W which are images of some vectors in V, that is. Handbook ofNEURAL NETWORK SIGNAL PROCESSING© 2002 by CRC Press LLC THE ELECTRICAL ENGINEERING AND APPLIED SIGNAL P. Let P n(x) be the space of polynomials in x of degree less than or equal to n, and consider the derivative operator d dx. Let $$T:V\rightarrow W$$ be a linear transformation where $$V$$ and $$W$$ be vector spaces with scalars coming from the same field $$\mathbb{F}$$. Similarly, we say a linear transformation T: max 1 i n di +2 r 2R2 n (p 2+ln r 1 )) 1 n+1 where the support of the distribution D is assumed to be contained in a ball of radius R. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. The kernel of L is the solution set of the homogeneous linear equation L(x) = 0. In fact, 4x+ 2y= 2(2x+ y) so those are the same equation which is equivalent to y= -2x. (b)The kernel of a linear transformation is a subspace of the domain. Trying to use matrices and matrix methods is almost a waste of time in this problem. Choose a simple yet non-trivial linear transformation with a non-trivial kernel and verify the above claim for the transformation you choose. The range of A is the columns space of A. The algorithm: The idea behind kernelml is simple. The most common form of radial basis function is a Gaussian distribution, calculated as:. visualize what the particular transformation is doing. Define pre-image of U, denoted T -1. If T(~x) = A~x, then the kernel of T is also called the kernel of A. Let be a linear transformation. Find the rank and nullity of a linear transformation from R^3 to R^2. 4 Linear Transformations The operations \+" and \" provide a linear structure on vector space V. F respects linear combinations, + q [F (VIC) of the following hold: i. Question: Why is a linear transformation called “linear”?. suppose T(x,y,z) = ( 2x-3y, x+4y-z, -x-7y+5z ) be a linear transformation. Similarly, we say a linear transformation T: max 1 i n di +2 r 2R2 n (p 2+ln r 1 )) 1 n+1 where the support of the distribution D is assumed to be contained in a ball of radius R. SPECIFY THE VECTOR SPACES Please select the appropriate values from the popup menus, then click on the "Submit" button. The dimension of the kernel of T is the same as the dimension of its null space and is called the nullity of the transformation. Find more Mathematics widgets in Wolfram|Alpha. Let $$T:V\rightarrow W$$ be a linear transformation where $$V$$ and $$W$$ be vector spaces with scalars coming from the same field $$\mathbb{F}$$. 2) When there is a large dataset linear SVM takes lesser time to train and predict compared to a Kernelized SVM in the expanded feature space. We solve by finding the corresponding 2 x 3 matrix A, and find its null space and column span. [Solution] To get an orthonormal basis of W, we use Gram-Schmidt process for v1 and v2. Similarly, a vector v is in the kernel of a linear transformation T if and only if T(v)=0. Griti is a learning community for students by students. (If all real numbers are solutions, enter REALS. $$Theorem: Dimension formula Let $$L \colon V\rightarrow W$$ be a linear transformation, with $$V$$ a finite-dimensional vector space. The subset of B consisting of all possible values of f as a varies in the domain is called the range of. Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Sx— 15y +4z x. , the solutions of the equation A~x = ~0. The spline can also be used for prediction. Find the kernel of the linear transformation L: V→W. The Polynomial kernel is a non-stationary kernel. The Matrix of a Linear Transformation We have seen that any matrix transformation x Ax is a linear transformation. These functions can be different types. Most off-the-shelf classifiers allow the user to specify one of three popular kernels: the polynomial, radial basis function, and sigmoid kernel. Null space. Griti is a learning community for students by students. Define the transformation \Omega: L(V,W) \to M_{m \times n} (\mathbb{R}) Stack Exchange Network Stack Exchange network consists of 176 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Let L : V →W be a linear transformation. Demonstrate: A mapping between two sets L: V !W. 2 (The Kernel and Range)/3. For example: random forests theoretically use feature selection but effectively may not, support vector machines use L2 regularization etc. The kernel of T, denoted by ker(T), is the set of all vectors x in Rn such that T(x) = Ax = 0. Q2 The Dimension of The Image and Kernel of a Linear Transformation 50 Points Q2. Determine whether the following functions are linear transformations. Before we look at some examples of the null spaces of linear transformations, we will first establish that the null space can never be equal to the empty set, in. Discuss this quiz (Key: correct, incorrect, partially correct. Then (1) is a subspace of. To help the students develop the ability to solve problems using linear algebra. Create a system of equations from the vector equation. Use the parameter update history in a machine learning model to decide how to update the next parameter set. Because Tis one-to-one, the dimension of the image of Tmust be n. You should think about something called the null space. Define by Observe that. We describe the range by giving its basis. The kernel of T, also called the null space of T, is the inverse image of the zero vector, 0, of W, ker(T) = T 1(0) = fv 2VjTv = 0g: It's sometimes denoted N(T) for null space of T. SPECIFY THE VECTOR SPACES Please select the appropriate values from the popup menus, then click on the "Submit" button. We define the kernel of $\phi$ to be. Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Sx— 15y +4z x. THE PROPERTIES OF DETERMINANTS a. Recall that if a set of vectors v 1;v 2;:::;v n is linearly independent, that means that the linear combination c. The kernel of a linear transformation {eq}L: V\rightarrow V {/eq} is the set of all polynomials such that {eq}L(p(t))=0 {/eq} Here, {eq}p(t) {/eq} is a polynomial. Image of a subset under a transformation. A linear map L∶V → W is called a linear isomorphism if ker(L) = 0 and L(V) = W. In both cases, the kernel is the set of solutions of the corresponding homogeneous linear equations, AX = 0 or BX = 0. It is one-one if its kernel is just the zero vector, and it is. 6, -1 ,-3-3 , 3 ,-1. Discuss this quiz (Key: correct, incorrect, partially correct. RHS of equation is a 2 row by 1 column matrix. We build thousands of video walkthroughs for your college courses taught by student experts who got an A+. Theorem If the linear equation L(x) = b is solvable then the. Hello and welcome back to Educator. The nullspace of a linear operator A is N(A) = {x ∈ X: Ax = 0}. We write ker(A) or ker(T). And we saw that earlier in the video. Let A = 2 4 0. In both cases, the kernel is the set of solutions of the corresponding homogeneous linear equations, AX = 0 or BX = 0. (7 pt total) Linear Transformations. Shed the societal and cultural narratives holding you back and let free step-by-step Linear Algebra: A Modern Introduction textbook solutions reorient your old paradigms. 1 Linear Transformations A function is a rule that assigns a value from a set B for each element in a set A. Transformation Matrices. (If all real numbers are solutions, enter REALS. There entires in these lists are arguable. Preimage and kernel example. 2 The Kernel and Range of a Linear Transformation4. More on matrix addition and scalar multiplication. as in Deﬁnition 1. Namely, linear transformation matrix learned in the high dimensional feature space can more appropriately map samples into their class labels and has more powerful discriminating ability. Next, we find the range of T. Use the kernel and image to determine if a linear transformation is one to one or onto. T [x, y, z, w] = [x + 2y + z - w] [2x + 3y - z + w] LHS of equation is a 4 row by 1 column matrix. A linear transformation T: R n → R m has an inverse function if and only if its kernel contains just the zero vector and its range is its whole codomain. nan_euclidean_distances (X) Calculate the euclidean distances in the presence of missing values. Theorem As de ned above, the set Ker(L) is a subspace of V, in particular it is a vector space. Use automated training to quickly try a selection of model types, and then explore promising models interactively. For instance, sklearn's SVM implementation svm. One can row reduce A to the identity matrix. We build thousands of video walkthroughs for your college courses taught by student experts who got an A+. Affine transformations", you can find examples of the use of linear transformations, which can be defined as a mapping between two vector spaces that preserves linearity. The problem comes when finding the Kernel basis. Justify your answers. These solutions are not necessarily a vector space. ] all keywords, in any order at least one, that exact phrase parts of words whole words. To see why image relates to a linear transformation and a matrix, see the article on linear. The $$\textit{nullity}$$ of a linear transformation is the dimension of the kernel, written$$ nul L=\dim \ker L. Ker(T) is the solution space to [T]x= 0. [Linear Algebra] Finding the kernel of a linear transformation. ANSWER Let p = ax2 +bx +c. Thus matrix multiplication provides a wealth of examples of linear transformations between real vector spaces. Anyway, hopefully you found that reasonably. (b)Find a linear transformation whose kernel is Sand whose range is S?. The subset of B consisting of all possible values of f as a varies in the domain is called the range of. 2 The kernel and range of a linear transformation. This theorem implies that every linear transformation is also a matrix transformation. Let P n(x) be the space of polynomials in x of degree less than or equal to n, and consider the derivative operator d dx. S: ℝ3 → ℝ3. com and welcome back to linear algebra. manhattan_distances (X[, Y, …]) Compute the L1 distances between the vectors in X and Y. The image of a linear transformation contains 0 and is closed under addition and scalar multiplication. Let be a linear transformation. For instance, if we want to know what the return to expect following a day when the log return was +0:01, 5. (some people call this the nullspace of L). We will see in the next subsection that the opposite is true: every linear transformation is a matrix transformation; we just haven't computed its matrix yet. Gaussian Kernel always provides a value between 0 and 1. Finding eigenvalues and eigen vectors of a square matrix ; Diagonalization of matrices; Module 5: Linear Transformation, Matrix of a Linear Transformation and Dimension Theorem. Similarly, we say a linear transformation T: mthen there exists inﬁnite solutions. AND LINEAR TRANSFORMATIONS Find the volume of the parallelepiped with one vertex at the origin and adjacent vertices are Find a polynomial p in P2 that spans the kernel of T, and describe the range of T. Then for any x ∞ V we have x = Íxáeá, and hence T(x) = T(Íxáeá) = ÍxáT(eá). 0004 From the previous lesson, we left it off defining what the range of a linear map is. The equationof-state formulation is based on the monotoric strain-hardening rule app1ied to the primarymore ». In this section, you will learn most commonly used non-linear regression and how to transform them into linear regression. Consider the linear system x+2y +3z = 1 2xy +2z =9 1. Note: Because Rn is a "larger" set than Rm when m < n, it should not be possible to map Rn to Rm in a one-to-one fashion. can be impractical to use. 1 2 -3 : 1/ 5 y 1 0 0 0 : - 7/. THE KERNEL IS A SUBSPACE: Let L : V !W be a linear transformation. Support vector machine with a polynomial kernel can generate a non-linear decision boundary using those polynomial features. Thus, the kernel of a matrix transformation T(x)=Ax is the null space of A.
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# Polynomial Division Problem

Polynomial Division Problem

Calculate the following and state the remainder:

2x^3+x^2-22x+20 divided by 2x-3

Thanks!

#### topsquark

Math Team
Polynomial Division Problem

Calculate the following and state the remainder:

2x^3+x^2-22x+20 divided by 2x-3

Thanks!
How are you to do this? You could use long division or sythnetic division.

What have you been able to work out?

-Dan

I calculated using long division and got the following:

x^2+2x-8 remainder -4

This question is from textbook and solution used synthetic division, but answer is different than mine...

Textbook solution gives: x^2+2x-8 remainder -2

#### skeeter

Math Team
$\dfrac{2x^3 +x^2-22x+20}{2x-3} = \dfrac{x^3 + \frac{x^2}{2} - 11x + 10}{x-\frac{3}{2}}$

synthetic division for the right side ...

Code:
[3/2]    1     1/2     -11    10
3/2        3    -12
------------------------------
1      2       -8   | -2
$\dfrac{x^3 + \frac{x^2}{2} - 11x + 10}{x-\frac{3}{2}} = x^2+2x-8 - \dfrac{2}{x-\frac{3}{2}} = x^2 + 2x - 8 - \dfrac{4}{2x-3}$

2 people

@ skeeter

Thanks, I see that (-2)/(x-3/2) is equivalent to (-4)/(2x-3)

So is the textbook incorrect? Should the remainder be (-2)/(x-3/2) instead of just -2?

Also, how to solve using long division then?

Here's txt solution using synthetic division:

https://imgur.com/BF0ZgTH

Last edited:

#### skeeter

Math Team
Code:
              x^2 + 2x  - 8
--------------------------
2x - 3 |  2x^3 + x^2 - 22x + 20
2x^3 - 3x^2
--------------------
4x^2 - 22x + 20
4x^2 -  6x
---------------
-16x + 20
-16x + 24
----------
-4
according to the remainder theorem, the remainder is -4, because ...

$f(x) = 2x^3 + x^2 - 22x + 20 \implies f\left(\frac{3}{2}\right) = -4$

Last edited:
2 people

@ Skeeter

Okay, so textbook is incorrect. Thanks for clarifying!

:-D

But I still don't understand how synthetic division gave incorrect remainder? Doesn't the -2 at the end of your synthetic division represent the remainder?

Last edited:

#### skeeter

Math Team
-2 is the remainder for $g(x) = x^2 + \dfrac{x^2}{2} - 11x + 10 \, \text{ since }\, g\left(\frac{3}{2}\right) = -2$

$2 \cdot g(x) = f(x) \implies$ the remainder for g(x) is half that for f(x).

It's all a matter of which polynomial is used to determine the remainder ... I support -4 as the remainder since f(x) divided by (2x-3) was the original problem.


# difference between quotient rule and product rule

Product rule :

$$\frac{d}{dx} \big(f(x)\cdot g(x)\big)=f'(x)\cdot g(x)+f(x)\cdot g' (x)$$

Quotient rule :

$$\frac{d}{dx} \frac{f(x)}{g(x)}=\frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{[g(x)]^2}$$

Suppose, the following is given in question. $$y=\frac{2x^3+4x^2+2}{3x^2+2x^3}$$

Simply, this is looking like Quotient rule. But, if I follow arrange the equation following way

$$y=(2x^3+4x^2+2)(3x^2+2x^3)^{-1}$$

Then, we can solve it using Product rule. As I was solving earlier problems in a pdf book using Product rule. I think both answers are correct. But, my question is, How does a Physicist and Mathematician solve this type question? Even, is it OK to use Product rule instead of Quotient rule in University and Real Life?

• Your just deriving the quotient rule on the fly, rather than assuming it is true and then applying it directly; there is nothing wrong with doing the former.
– user785085
May 8, 2021 at 13:39
• First thing a mathematician would do is write $$y=1+\frac{x^2+2}{2x^3+3x^2}$$ before taking the derivative. May 8, 2021 at 16:14
• What do you mean "I think both answers are correct."? If you do the math properly, you'll get the same answer with both methods. May 9, 2021 at 0:10
• Real physicists and mathematicians stick the whole thing into their computer algebra system and don't worry about the precise algorithms it follows, unless they have a particular reason to risk silly errors by doing that kind of calculation by hand. Symbolic differentiation is a solved problem; one doesn't earn any "purity points" in the real world by doing it the hard way. May 9, 2021 at 0:33
• @JosephSible-ReinstateMonica Yes! I got same answer :)
– user876873
May 9, 2021 at 4:30

Note that $$((g(x))^{-1})'=-g'(x)(g(x))^{-2}$$. Then, applying the product rule: $$\left(\frac{f(x)}{g(x)}\right)'=\left(f(x)\cdot \frac{1}{g(x)}\right)'=\frac{f'(x)}{g(x)}+\frac{-g'(x)f(x)}{(g(x))^2}=\frac{f'(x)g(x)-f(x)g'(x)}{(g(x))^2}$$ which is the quotient rule

• That's very helpful. But, it was my main question How does a Physicist and Mathematician solve this type question? Even, is it OK to use Product rule instead of Quotient rule in University and Real Life?
– user876873
May 8, 2021 at 13:44
• @Istiak Please don't keep making that bold. Sure, it's OK, because any valid proof is OK; there are no "rules" beyond that. To be honest, I hardly ever see mathematicians or physicists explicitly using the quotient rule. It seems to be more for teaching than anything else.
– J.G.
May 8, 2021 at 13:45
• I'd always use the quotient rule on a quotient because it is usually much simpler to work with $g'$ than $(\frac1g)'$ May 8, 2021 at 13:45
• @J.G. Actually, I had read in a meta post that is saying,you should bold when it may need to be attracted or, something just like this. I don't remember. That's why I was just making that bold
– user876873
May 8, 2021 at 13:48
• @Istiak I see, cool. Thanks, though, for your edit that means you now only bold it in the question.
– J.G.
May 8, 2021 at 13:52

How does a Physicist and Mathematician solve this type question? Even, is it OK to use Product rule instead of Quotient rule in University and Real Life?

is that any experienced scientist knows several methods to solve problems and uses those that are most convenient for them at that particular time.

I would look at that derivative and use the quotient rule. But if there was something in the source of the problem that suggested that it made more sense to write the denominator as $$(3x^2+2x^3)^{-1}$$ then the product rule would be more appropriate.

the quotient rule is not a separate and non-compatible rule. It is just the product rule inserting $$\frac 1 g$$ instead of $$g$$.

let's see...

to make it clear... I show you, prove it for you, how quotient rule is just compatible with product law.

$$\left(\frac f g\right)'=\left(f\cdot \frac 1 g\right)'$$

as product law says:

$$=f' \cdot \frac 1 g +f\cdot \left (\frac 1 g\right)'$$

while $$\displaystyle\left(\frac 1 g \right)' = \frac {-g'}{g^2}$$, we would have

$$=f'\cdot \frac 1 g+f\cdot \frac{-g'}{g^2}=\frac{f'\cdot g-f\cdot g'}{g^2}.$$

Note:

You may know that $$\displaystyle\left(\frac 1 h \right)' = \frac {-h'}{h^2}$$ could be calculated by product rule, as if one consider the product $$\displaystyle\left(\frac 1 h \cdot h \right) = 1$$, and calculate the derivative of both sides of the equation. one the left hand side we have a constant which may already know the derivative is $$0$$, but on the other side we see a product so by applying the product rule we have $$0=h' \cdot \frac 1 h + h\cdot \left (\frac 1 h \right)'.$$

Therefore $$h\cdot \left (\frac 1 h \right)'=-h' \cdot \frac 1 h.$$

And so $$\left (\frac 1 h \right)'=-h' \cdot \frac 1 {h^2} = -\frac {h'} {h^2}.$$

• $g^{-1}$ is a poor choice of notation here, as it usually means the inverse of $g$, and not the reciprocal. May 9, 2021 at 0:19
• @XanderHenderson I fixed the notation, thanks... as you remember I never used the $g^{-1}$ notation in the followings May 6 at 15:12


# What is wrong with the argument $1 = \lim_{n\to \infty} n/n = \lim_{n\to\infty} (1/n+1/n+\dotsb+1/n) = 0$?

Let we have

$$\begin{equation*} n\times\frac{1}{n}=\frac{1}{n}+\frac{1}{n}+\cdots+\frac{1}{n}\mbox{ (n times)}. \end{equation*}$$

Taking $$\lim_{n\to\infty}$$ to both sides, we get

$$\begin{eqnarray*} \lim_{n\to\infty}1 &=& \lim_{n\to\infty}\left(\frac{1}{n}+\frac{1}{n}+\cdots+\frac{1}{n}\right)\\ \Longrightarrow1&=&0+0+\cdots+0\mbox{ (n times)}\\ &=&0. \end{eqnarray*}$$

I am not an expert of math, and confused that where the confusion is.

• An informal answer here: The problem is the "dot...dot...dot" part. There is such thing as an indeterminate form, being zero times infinity. And that is typically NOT zero. With the "dot...dot...dot" notation, you are "reading" the problem as a finite amount of $1/n$ terms (even though it is not your intention) and hence you come up with zero as the (incorrect) answer. – imranfat Aug 5 at 2:17
• Also, I up-voted the question because it is a good question. You pose this question to introductory calculus students and I bet half of them cannot point out the issue. – imranfat Aug 5 at 2:19
• By your logic, on the left you also have $\lim n \cdot \frac{1}{n} = \lim_{n\to\infty} n \cdot \lim_{n\to\infty} \frac{1}{n} = \infty \cdot 0.$ – mjw Aug 5 at 2:20
• The problem is that you can't only take limit on $n$ only on part of the expression. There is the "$n$ times" which also depended on the $n$ you are taking to $\infty$. And no, you can't split the limits up in two in general. – user10354138 Aug 5 at 2:21
• – Angina Seng Aug 5 at 2:27

This is an excellent argument that we cannot in general find a limit by taking the limits of the parts of an expression.

When many students are first introduced to limit laws, they see their instructor go through a lot of complicated math in order to prove things that feel obvious. In this case, the relevant one is the addition law:

$$\lim_{x \to c}\left(f(x) + g(x)\right) = \lim_{x\to c}f(x) + \lim_{x\to c}g(x)$$

This seems obvious, right? A limit means "what number does this expression get close to". Of course $$f(x) + g(x)$$ would get close to the sum of whatever $$f(x)$$ gets close to and whatever $$g(x)$$ gets close to. So why does the instructor (or the textbook) spend half a page messing around with $$\epsilon$$s and $$\delta$$s to prove the law?

The answer is because of exactly the sort of thing you've pointed out. There are situations where the "intuitive" approach to limits stops working, essentially because infinity is hard. For those situations, we need to rely on the proof. Crucially, in this case, the proof relies on there being only two things added together. This means, if we want to adhere perfectly to the law as stated, we have to jump through hoops like this:

\begin{align*} \lim_{x \to c} \left(f(x) + g(x) + h(x)\right) &= \lim_{x \to c} \left(\left(f(x) + g(x)\right) + h(x)\right)\\ &= \lim_{x \to c}\left(f(x) + g(x)\right) + \lim_{x \to c}h(x)\\ &= \lim_{x \to c}f(x) + \lim_{x \to c}g(x) + \lim_{x \to c}h(x) \end{align*}

We can do the same to deal with four, or five, or five hundred things added together. But how would we deal with $$n$$ things added together, when $$n$$ changes over the course of the limit? If we "peel off" one like I did above, there'd still be infinitely many left over. In other words, even with aggressive uses of this limit law, we can only handle sums of fixed size. One that "grows", like $$\frac1n + \frac1n + \cdots + \frac1n$$ does, can't be handled this way.

To summarize: Many of the limit laws feel like they're just saying "take the limit of the parts of the expression". This isn't true; in fact, they're saying "here is one precise way in which you can find a limit by using the limits of the parts". If you want to do something to a limit that isn't one of the standard limit laws, you're doing something special, which means you'll need to go back to the definition of the limit (or something similar) in order to make sure that what you're doing works.

• Excellent answer. I don't know if this is worth noting, but we could peel off one term (or finitely many) from the expression in question - we just are always left with a sum that doesn't have a fixed size. e.g. $\displaystyle{\lim_{n\to\infty}}\underbrace{\frac1n+\frac1n+\cdots}_{n\text{ terms}}=\displaystyle{\lim_{n\to\infty}}\frac1n+\displaystyle{\lim_{n\to\infty}}\underbrace{\frac1n+\frac1n+\cdots}_{n-1\text{ terms}}$ – Mark S. Aug 5 at 13:05


# Checking if a symbolic symmetrical Matrix is negative definite

I have the following problem finding the value ranges for the parameters of a symbolic symmetrical matrix in order to make it negative definite:

The matrix I'm talking about looks as follows

A := {{-1, -b, a, 0}, {-b, -1, 0, 0}, {a, 0, -a, -b a}, {0, 0, -b a, -a}}

and as you can see in matrix form, it is a symmetrical matrix

\begin{array}{cccc} -1 & -b & a & 0 \\ -b & -1 & 0 & 0 \\ a & 0 & -a & -a b \\ 0 & 0 & -a b & -a \\ \end{array}

Now I'm trying to find the value ranges of a and b in order to make the matrix negative definite. It is important that b depends on a and not the other way round, since the matrix is part of an economic model, which doesn't make any sense otherwise.

First I used the approach to find the value ranges, which make all Eigenvalues negative and thus lead to a negative definite matrix

Reduce[Eigenvalues[A] < 0, {a, b}]

which yields

0 < a < 1 && -Sqrt[1 - Sqrt[a]] < b < Sqrt[1 - Sqrt[a]]

Everything fine so far. But then I tried a different approach. If the k-th order leading principal minor of the matrix has sign (-1)^k, then the matrix should be negative definite, so I'm expecting the same result:

A1 := {{-1}}
A2 := {{-1, -b}, {-b, -1}}
A3 := {{-1, -b, a}, {-b, -1, 0}, {a, 0, -a}}

Reduce[{Det[A1] < 0, Det[A2] > 0, Det[A3] < 0, Det[A] > 0}, {a, b}]

which yields

0 < a < 1 && Root[1 - a - 2 #1^2 + #1^4 &, 2] < b < Root[1 - a - 2 #1^2 + #1^4 &, 3]

ToRadicals[
0 < a < 1 && Root[1 - a - 2 #1^2 + #1^4 &, 2] < b < Root[1 - a - 2 #1^2 + #1^4 &, 3]]

0 < a < 1 && -Sqrt[1 - Sqrt[a]] < b < Sqrt[1 + Sqrt[a]]

As you can see, the result is different than in the first approach (to be more specific the upper bound of b is different), which makes no sense, since both approaches should yield the same result.

Does anyone know what I am doing wrong or which of the results is correct?

Thanks a lot,

Phil

• The latter answer is incorrect: A /. {a -> 1/2, b -> Sqrt[1 + Sqrt[1/2]]} // Eigenvalues // N has a positive eigenvalue. Maybe you set up the deterimnants incorrectly since it looks like a sign error. – bill s Aug 11 '17 at 18:51
• See @Szabolcs' answer to a question about reordering when using ToRadicals on Root objects. – Carl Woll Aug 11 '17 at 19:27
• Thanks @CarlWoll, I think the ordering of the Root objects is on the right track. But I still haven't found a solution for my specific case – PTSammy Aug 13 '17 at 12:56

Both of your approaches yield the same answer. it is the application of ToRadicals that causes the answer to be different. First, compare the two limits before the application of ToRadicals:

upperLimit1 = Sqrt[1 - Sqrt[a]];
upperLimit2 = Root[1 - a - 2 #1^2 + #1^4 &, 3];

Plot[upperLimit1, {a, 0, 1}]
Plot[upperLimit2, {a, 0, 1}]

upperLimit1 and upperLimit2 are the same over the region 0 < a < 1. Converting the Root object into radicals is problematic because the Root ordering depends on the parameter a. One suggestion would be to not use ToRadicals and just work with the Root objects. If you really want radicals, a naive application of ToRadicals:

ToRadicals[upperLimit2]

Sqrt[1 + Sqrt[a]]

is only correct for some values of the parameter a. However, in your case, you know something about the parameter a, so you should make use of that by giving ToRadicals an assumption:

ToRadicals[upperLimit2, Assumptions -> 0 < a < 1]

Sqrt[1 - Sqrt[a]]

Note that using:

ToRadicals[0 < a < 1 && upperLimit2]

0 < a < 1 && Sqrt[1 + Sqrt[a]]

does not cause ToRadicals to use 0 < a < 1 as an assumption. The assumption needs to be given explicitly as an option to ToRadicals.

In this answer I will briefly go through the method of using the leading principal minors to derive negative definiteness. Hope this helps you verify your results

A matrix is negative definite when ALL its leading principal minors alternate in sign, with the k-th order leading principal minor having the same sign as $(-1)^k$.

Please note, I am NOT talking about a problem instance with constraints ie I am not talking about bordered matrices. In that case, the rules are a bit different, but this is not the case here.

Also, note, that the order of a minor is derived from the number of columns and rows one has to delete in order to obtain the corresponding submatrix.

Counting is straightforward: For an $nxn$ matrix the k-th order leading principal minor is produced by deleting the last $n-k$ rows and columns of the original matrix. In this fashion, the 1-st order leading principal minor of a $4x4$ matrix is produced after deleting the last $4-1=3$ rows and columns while the 2-nd order leading principal minor of the same matrix needs to have the last $4-2=2$ rows and columns deleted etc.

Finally, note that taking the n-th order leading principal minor of an $nxn$ matrix means deleting $n-n=0$ rows and columns ie the n-th order minor is the determinant of the matrix itself.

Mathematica has (arguably) a lot of ways to produce the leading principal submatrices that are needed to produce the minors (their determinants). One simple and fast way to do it, is:

LeadingPrincipalMinors=Array[Minors[A, #][[1, 1]] &, 4]

Now, checking to verify if the signs of the minors are in agreement with the sign rule we can do the following:

Reduce[
And @@ MapIndexed[(
Reduce[#1 (-1)^First[#2] > 0, b[a], Reals]

The 'trick' I use in this piece of code is to take advantage of the fact that, when two numbers have the same sign, their product is positive. This is what the following excerpt of code from above, does:

#1 (-1)^First[#2] >= 0

Please, note that I have used b[a] instead of b in order to make explicit the dependence of b on a. If you chose to do so, you have to replace the definition of matrix A with something like

A=A/.b->b[a]

but that is not necessary. I just find it useful to have dependence relationships be defined as explicitly as possible in my code.

Therefore the range of values over which the initial matrix is negative definite depends on a and b[a] being in an appropriate range of values.

Hope that helps :)

## -- update --

executing

Reduce[
And @@ MapIndexed[(
Reduce[#1 (-1)^First[#2] > 0, b[a], Reals]

and

Reduce[And @@ Thread[Eigenvalues[A] < 0], b[a], Reals]

yields the same output, namely:

0 < a < 1 && -Sqrt[1 - Sqrt[a]] < b[a] < Sqrt[1 - Sqrt[a]]

• Thanks for your answer! There is a small mistake in your code #1 (-1)^First[#2] >= 0 needs to be #1 (-1)^First[#2] > 0, otherwise the leading principal minors can also be 0, which does not imply negative definiteness. Changing the code this way leads to the same result I had earlier in my second approach and does not solve my problem – PTSammy Aug 13 '17 at 13:00
• @PTSammy you are right; I wrote in the body of the text "[w]hen two numbers have the same sign, their product is positive" but miss-typed ">=" instead of the correct ">" i the code segment. Will correct it. – user42582 Aug 13 '17 at 15:15
• @PTSammy As far as the last part of your comment is concerned-just for the sake of argument-technically speaking, my answer, answers the second part of your question "[D]oes anyone know what I am doing wrong or which of the results is correct?" as I already stated it would do, in the beginning of the text. Having said that, it is true that it does not identify what your possible error might be-which is something which I didn't state it would do. – user42582 Aug 13 '17 at 15:22
• You're right, you didn't state that. No worries, I didn't mean to offend you ;) – PTSammy Aug 13 '17 at 18:03
• @PTSammy all's well; hope you figure it out; have a nice time :) – user42582 Aug 13 '17 at 18:21


To show that $\ell^\infty$ is a vector space does it suffice to show it satisfies the Subspace Criteria?

If I want to show that $\ell^\infty$ (the set of all bounded sequences of real numbers) is a vector space does it suffice to show that since $\ell^\infty \subset \mathbb{R}_\infty$ (the set of all sequences of real numbers) and $\mathbb{R}_\infty$ is a vector space that the following are true:

(a.) $0\in \ell^\infty$ (this is trivial because the 0 sequence is bounded and so is in $\ell^\infty$)

(b.) For $u,v \in \ell^\infty$ we have that $u+v \in \ell^\infty$.

We can show this from the definition of bounded sequences, $u = \left\{x_1,x_2,\dots\right\}$ is bounded if $\exists M>0$ such that $|x_n|<M$ for all $N\in\mathbb{N}$. Similarly for $v = \left\{y_1,y_2,\dots\right\}$ bounded by $T>0$. Therefore:

$$|(u+v)_n|=|x_n +y_n| \leq |x_n| + |y_n| < M+T$$

Therefore for any two bounded sequences $u,v$ the sum $u+v$ is also bounded and so $\ell^\infty$ is closed under addition.

(c.) A similar proof shows that $c\cdot u \in \ell^\infty$ for all $c \in \mathbb{R}$

This shows that $\ell^\infty$ satisfies all the necessary conditions to be a subspace of $\mathbb{R}_\infty$. Therefore I think I have shown $\ell^\infty$ is a vector space, but then the problem has me show that $\mathbb{R}^\infty$ (the set of sequences that are eventually zero) is a subspace of $\ell^\infty$ and so I think I may have messed up.

• Your proof looks essentially correct to me. I think you may mean $|x_n + y_n| \le |x_n| + |y_n|$ in the body of your proof. Dec 6 '16 at 3:23
• Thank you, I copied the Latex from my homework so thanks a ton for pointing it out to me. Dec 6 '16 at 3:24

Your method is perfectly correct. The same criteria can be used to show that $\mathbb R^\infty$ is a subspace of $\ell^\infty$.
• Awesome. I took a walk and considered the proof for $\mathbb{R}^\infty$, since it's a finite sequence there exists a natural number $N \in \mathbb{N}$ such that for $n > N$ we have that $x_n = 0$ therefore we can take the upper bound to be $\max x_n$ and go about the same procedure to prove it is a subspace of $\ell^\infty$. Does this seem right (with more attention to detail added when I write the proof, of course) Dec 6 '16 at 3:28