Area of a square. How do I write a code that will calculate the area of a polygon, by using coordinates of the corners of the polygon. Polygon Calculator. Area. Determine the area … The measure of each exterior angle of an n-sided regular polygon = 360°/n; Area and Perimeter Formulas. One hectare is about $$\text{0,01}$$ square kilometres and one acre is about $$\text{0,004}$$ square kilometres. The area that wasn’t subtracted (grey) is the area of the polygon! Please help!!!! $$\therefore$$ Area occupied by square photo frame is $$25$$ sq. Area of a circular sector. Introduction to Video: Area of Regular Polygons; 00:00:39 – Formulas for finding Central Angles, Apothems, and Polygon Areas; Exclusive Content for Member’s Only ; 00:11:33 – How to find the … Types of Polygons Regular or Irregular. If two adjacent points along the polygon’s edges have coordinates (x1, y1) and (x2, y2) as shown in the picture on the right, then the area (shown in blue) of that side’s trapezoid is given by: The Algorithm – Area of Polygon. Once done, open the attribute table to see the result. Regular: Irregular: The Example Polygon. They assume you know how many sides the polygon has. 3. In a triangle, the long leg is times as long as the short leg, so that gives a length of 10. is twice that, or 20, and thus the perimeter is six times that or 120. Chapter 13: Measurements. Validation. Area of a parallelogram given base and height. Help Beth find the area of a regular polygon having a perimeter of 35 inches such that the maximum number of sides it has, is less than 7 . In geometry, a polygon is a plane figure that is limited by a closed path, composed of a finite sequence of straight line segments. The area of the polygon is Area = a x p / 2, or 8.66 multiplied by 60 divided by 2. End of chapter exercises. person_outline Timur … To see how this equation is derived, see Derivation of regular polygon area formula. where, S is the length of any side N is the number of sides π is PI, approximately 3.142 NOTE: The area of a polygon that has infinite sides is the same as the area a circle. Area of a cyclic quadrilateral. See our Version 4 Migration Guide for information about how to upgrade. Link × Direct link to this answer. Hint is I will have to use the cosine law????? Sign in to comment. You need the perimeter, and to get that you need to use the fact that triangle OMH is a triangle (you deduce that by noticing that angle OHG makes up a sixth of the way around point H and is thus a sixth of 360 degrees, or 60 degrees; and then that angle OHM is half of that, or 30 degrees). Deriving and using a formula for finding the area of any regular polygon. Yes. = | 1/2 [ (x 1 y 2 + x 2 y 3 + … + x n-1 y n + x n y 1) –. To find the area of a regular polygon, all you have to do is follow this simple formula: area = 1/2 x perimeter x apothem. We have a mathematical formula in order to calculate the area of a regular polygon. The method used when evaluating a feature's area or perimeter. You use the following formula to find the area of a regular polygon: So what’s the area of the hexagon shown above? Area is always a positive number. Area of a Polygon. A polygon is a two-dimensional shape that is bounded by line segments. Polygon area. Qwertie. First, you have this part that's kind of rectangular, or it is rectangular, this part right over here. They assume you know how many sides the polygon has. The area of any regular polygon is equal to half of the product of the perimeter and the apothem. Types of polygon. Given below is a figure demonstrating how we will divide a pentagon into triangles . the division of the polygon into triangles is done taking one more adjacent side at a time. This tutorial will cover creating a polygon on the map and computing/printing out to the console information such as area, perimeter, etc about the polygon. Then, find the area of the irregular polygon. The area formula is derived by taking each edge AB and calculating the (signed) area of triangle ABO with a vertex at the origin O, by taking the cross-product (which gives the area of a parallelogram) and dividing by 2. Is it a Polygon? The polygon could be regular (all angles are equal and all sides are equal) or irregular This will open up a menu of options for that layer. Area of the polygon = $$\dfrac{4 \times 5 \times 2.5}{2} = 25$$ sq. Poly-means "many" and -gon means "angle". (x 2 y 1 + x 3 y 2 + … + x n y n-1 + x 1 y n) ] |. How to Calculate the Area of Polygon in ArcMap. Regular polygon calculator is an online tool to calculate the various properties of a polygon. We then find the areas of each of these triangles and sum up their areas. Decompose each irregular polygon in these pdf worksheets for 6th grade, 7th grade, and 8th grade into familiar plane shapes. Enter any 1 variable plus the number of sides or the polygon name. We then find the areas of each of these triangles and sum up their areas. Type. Next, select the polygon file that you want to calculate area on and right click. Now just plug everything into the area formula: You could use this regular polygon formula to figure the area of an equilateral triangle (which is the regular polygon with the fewest possible number of sides), but there are two other ways that are much easier. If the angles are all equal and all the sides are equal length it is a regular polygon. The formulas for areas of unlike polygon depends on their respective shapes. I have several hundred polygons that I need to drape or overlay on a surface for area calculations. To compute the area using the faster but less accurate spherical model use ST_Area(geog,false). 1 hr 23 min. Triangles, quadrilaterals, pentagons, and hexagons are all examples of polygons. Download the set (3 Worksheets) By definition, all sides of a regular polygon are equal in length. Interior angles of polygons. Let us learn here to find the area of all the polygons. Worksheet on Area of a Polygon is helpful to the students who are willing to solve the questions on area of the pentagon, square, hexagon, octagon, and n-sided polygons. 6. Let us discuss about area of polygon. A regular polygon is equilateral (it has equal sides) and equiangular (it has equal angles). Perimeter—Evaluates the length of the entire feature or its individual parts or segments. This approach can be used to find the area of any regular polygon. Perimeter: Perimeter of a polygon is the total distance covered by the sides of a polygon. but see Trigonometry Overview). Four different ways to calculate the area are given, with a formula for each. Area of Irregular Polygons. Area of Irregular Polygons Introduction. Determine the area of the trapezoid below. Polygon area An online calculator calculates a polygon area, given lengths of polygon sides and diagonals, which split polygon to non-overlapping triangles. Constraint. Let's use this polygon as an example: Coordinates. 0 Comments. The solution is an area of 259.8 units. The area and perimeter of different polygons are based on the sides. Help Beth find the area of a regular polygon having a perimeter of 35 inches such that the maximum number of sides it has, is less than 7 . It is always a two-dimensional plane. Area of a rhombus. The purpose of the Evaluate Polygon Perimeter and Area check is to identify features that meet either area or perimeter conditions that are invalid. Learn how to find the area of a regular polygon when only given the radius of the the polygon. And you don’t have to start at the top of the polygon — you can start anywhere, go all the way around, and the numbers will still add up to the same value. However, if the polygon is cyclic then the sides do determine the area. By definition, all sides of a regular polygon are equal in length. By Mark Ryan. Calculate from an regular 3-gon up to a regular 1000-gon. I just thought I would share with you a clever technique I once used to find the area of general polygons. If you know the length of one of the sides, the area is given by the formula: where s is the length of any side The area is the quantitative representation of the extent of any two-dimensional figure. It can be used to calculate the area of a regular polygon as well as various sided polygons such as 6 sided polygon, 11 sided polygon, or 20 sided shape, etc.It reduces the amount of time and efforts to find the area or any other property of a polygon. Finally learners investigate the effects of multiplying any dimension by a constant factor $$k$$. Calculating the area of a polygon can be as simple as finding the area of a regular triangle or as complicated as finding the area of an irregular eleven-sided shape. The above formula is derived by following the cross product of the vertices to get the Area of triangles formed in the polygon. Find the area and perimeter of the polygon. circle area Sc . Example 2 . That’s how it works. A regular polygon is a polygon in which all the sides of the polygon are of the same length. Most require a certain knowledge of trigonometry (not covered in this volume, but see Trigonometry Overview). 2. Find the area of any regular polygon by using special right triangles, trigonometric ratios (i.e., SOH-CAH-TOA), and the Pythagorean theorem. Polygon (straight sides) Not a Polygon (has a curve) Not a Polygon (open, not closed) Polygon comes from Greek. Central Angle of a Regular Polygon. Example of the Polygon Area Calculation. Sign in to answer this question. Area: Area is defined as the region covered by a polygon in a two-dimensional plane. Area of: rectangle | square | parallelogram | triangle | trapezoid | circle. Content covered in this chapter includes revision of volume and surface area for right-prisms and cylinders. 4. . Area of a rhombus. Note as well, there are no parenthesis in the "Area" equation, so 8.66 divided by 2 multiplied by 60, will give you the same result, just as 60 divided by 2 multiplied by 8.66 will give you the same result. Calculate from an regular 3-gon up to a regular 1000-gon. Polygon Calculator. This program calculates the area of a polygon, using Matlab. Area of a polygon (Coordinate Geometry) A method for finding the area of any polygon when the coordinates of its vertices are known. They were all drawn on a horizontal plane without taking into account the elevation changes of the terrain. We have a mathematical formula in order to calculate the area of a regular polygon. Radius of circle given area. This work is then extended to spheres, right pyramids and cones. This is because any simple n-gon ( having n sides ) can be considered to be made up of (n − 2) triangles, each of which has an angle sum of π radians or 180 degrees. For geometry types a 2D Cartesian (planar) area is computed, with units specified by the SRID. $$\therefore$$ Area occupied by square photo frame is $$25$$ sq. A central angle of a regular polygon is an angle whose vertex is the center and whose sides contain two consecutive vertices of the polygon. Just enter the coordinates. We generally use formulas to calculate areas. Rate me: Please Sign up or sign in to vote. Enter any 1 variable plus the number of sides or the polygon name. The coordinates of the vertices of this polygon are given. Use the appropriate area formula to find the area of each shape, add the areas to find the area of the irregular polygons. You can see how this works with triangle OHG in the figure above. Just as one requires length, base and height to find the area of a triangle. Area of a rectangle. Triangles, quadrilaterals, pentagons, and hexagons are all examples of polygons. If you want to recreate it you can find the source code here. Plot a polygon onto the map; Compute and print out information about the polygon; Dependencies. The Algorithm – Area of Polygon The idea here is to divide the entire polygon into triangles. Area of a triangle (Heron's formula) Area of a triangle given base and angles. Vote. Calculates side length, inradius (apothem), circumradius, area and perimeter. Polygon Calculator. We can compute the area of a polygon using the Shoelace formula . The length of the apothem is given. FAQ. Polygons are 2-dimensional shapes. The apothem of a regular polygon is a line segment from the center of the polygon to the midpoint of one of its sides. Area of a quadrilateral. Lesson Summary. A regular polygon is equilateral (it has equal sides) and equiangular (it has equal angles). Before we move further lets brushup old concepts for a better understanding of the concept that follows. Area of a regular polygon. Next. Area of a parallelogram given base and height. Polygons—Evaluates the area or perimeter of the entire polygon … Area of a regular polygon. The formulae below give the area of a regular polygon. For example, the following, self-crossing polygon has zero area: 1,0, 1,1, 0,0, 0,1 (If you want to calculate the area of the polygon without running into problems like negative area, and overlapping areas described below, you should use the polygon perimeter technique.) inches. Objectives. I am not sure how to do this. If DC = 1.9 cm, FE = 5.6 cm, AF = 4.8 cm, and BC = 10.9 cm, find the length of the other two sides. If two adjacent points along the polygon’s edges have coordinates (x1, y1) and (x2, y2) as shown in the picture on the right, then the area (shown in blue) of that side’s trapezoid is given by: area = (x2 - x1) * (y2 + y1) / 2. Overview. The area of any given polygon whether it a triangle, square, quadrilateral, rectangle, parallelogram or rhombus, hexagon or pentagon, is defined as the region occupied by it in a two-dimensional plane. You must supply the x and y coordinates of all vertices. Limitations This method will produce the wrong answer for self-intersecting polygons, where one side crosses over another, as shown on the right. Polygons A polygon is a plane shape with straight sides. Area of a circle. Area of a Rectangle A rectangle is … Calculating the area of a polygon. Show Hide all comments. First, open up an ArcGIS session and load in the polygon data you want to calculate the area on. To find the area of a regular polygon, you use an apothem — a segment that joins the polygon’s center to the midpoint of any side and that is perpendicular to that side (segment HM in the following figure is an apothem). Most require a certain knowledge of trigonometry (not covered in this volume, Now I have a new column called Area_calculation which contains total area for each polygon. Solution . For geography types by default area is determined on a spheroid with units in square meters. 13.1 Area of a polygon (EMA7K) Area. But, areas of are negative. Suppose, to find the area of the triangle, we have to know the length of its base and height. In case the students are preparing for any kind of test, then they can start preparation from this Area of the Polygon … Decompose each irregular polygon in these pdf worksheets for 6th grade, 7th grade, and 8th grade into familiar plane shapes. Access to Google Earth Engine’s Code Editor; Creating/Plotting a Polygon Polygon example. First, you can get the area of an equilateral triangle by just noting that it’s made up of two triangles. Area of a cyclic quadrilateral. Right prisms and cylinders. Write down the formula for finding the area of a regular polygon. Second, the equilateral triangle has its own area formula so that’s a really easy way to go assuming you’ve got some available space on your gray matter hard drive: Area of an equilateral triangle: Here’s the formula. New to … The acre and the hectare are two common measurements used for the area of land. I would like to determine the area of the raster (only category 1) within the polygon/shapefile . Area of Irregular Polygons Introduction. I just thought I would share with you a clever technique I once used to find the area of general polygons. Calculates side length, inradius (apothem), circumradius, area and perimeter. Use this calculator to calculate properties of a regular polygon. Next Success Install Microsoft SQL Server 2017 on Ubuntu Server 20.04 7 months ago. Area of regular polygon = … Area calculator See Polygon area calculator for a pre-programmed calculator that does the arithmetic for you. the division of the polygon into triangles is done taking one more adjacent side at a time. And the final formula, that computes the target polygon area:. A fast and simple algorithm. It represents the number of square units needed to cover a shape, such as a polygon or a circle. Use this calculator to calculate properties of a regular polygon. A polygon is any 2-dimensional shape formed with straight lines. Questionnaire. Area of the polygon = $$\dfrac{4 \times 5 \times 2.5}{2} = 25$$ sq. inches. Importantly, we’ve chosen a point for calculating the areas. Video – Lesson & Examples. For any two simple polygons of equal area, the Bolyai–Gerwien theorem asserts that the first can be cut into polygonal pieces which can be reassembled to form the second polygon. Then, find the area of the irregular polygon. Depending on the information that are given, different formulas can be used to determine the area of a polygon, below is a list of these formulas: Use the one that matches what you are given to start. Area of a parallelogram given sides and angle. Area of a parallelogram given sides and angle. As shown below, this means that we must find the perimeter (distance all the way around the hexagon) and the measure of the apothem using right triangles and trigonometry. Side of polygon given area. person_outlineTimurschedule 2011-06-06 07:13:58. 4.43/5 (3 votes) 4 Jun 2013 CPOL. The measure of any int… An online calculator calculates a polygon area, given lengths of polygon sides and diagonals, which split polygon to non-overlapping triangles. This program calculates the area of a polygon, using Matlab.You must supply the x and y coordinates of all vertices. It is measured in square units. Note: this page is part of the documentation for version 3 of Plotly.py, which is not the most recent version. I'm trying to write a code to calculate the area of a polygon, but it looks like something isn't adding up. So this irregular polygon has an area of 126 cm 2. Polygons A polygon is a plane shape with straight sides. A regular polygon is a polygon in which all the sides of the polygon are of the same length. To find the area of a regular hexagon, or any regular polygon, we use the formula that says Area = one-half the product of the apothem and perimeter. Area of an arch given angle. double The formulae below give the area of a regular polygon. The areas or formulas for areas of different types of polygondepends on their shapes. For instance, let’s take the polygon below and use the above formula to compute its area:. They are made of straight lines, and the shape is "closed" (all the lines connect up). Coordinates must be entered in order of successive vertices. inches. The lengths of the sides of a polygon do not in general determine its area. 1. Example 2 . Coordinates must be entered in order of successive vertices. A polygon is any 2-dimensional shape formed with straight lines. $\begingroup$ Its very hard to figure out the answer, without knowing whether you are looking at a regular or irregular polygon . The idea here is to divide the entire polygon into triangles. But I think that typing " derive area of polygon " in Google may fetch you lots of links. Make sure your data is in a projection system. Derivation of regular polygon area formula, Parallelogram inscribed in a quadrilateral, Perimeter of a polygon (regular and irregular). The number of square units it takes to completely fill a regular polygon. Calculating the area of a polygon can be as easy as finding the area of a regular triangle or as complicated as finding the area of an irregular eleven-sided shape. Watch and learn how to find the area of a regular polygon. In this program, we have to find the area of a polygon. As one wraps around the polygon, these triangles with positive and negative areas will overlap, and the areas between the origin and the polygon will be canceled out and … qgis tutorial. inches. (See also: Computer algorithm for finding the area of any polygon .) Use the one that matches what you are given to start. Given the length of a side. Calculates the side length and area of the regular polygon inscribed to a circle. Area of a trapezoid. You can calculate the area of a polygon by adding the areas of the trapezoids defined by the polygon’s edges dropped to the X-axis. Each section consists of a rectangle and a triangle. Polygons are 2-dimensional shapes. It looks like geojson.io is not calculating the area after projecting the spherical coordinates onto a plane like you are, but rather using a specific algorithm for calculating the area of a polygon on the surface of a sphere, directly from the WGS84 coordinates. 1. area ratio Sp/Sc Customer Voice. Irregular polygons are polygons that do not have equal sides or equal angles. polygon area Sp . All I'm looking for are areas, I'm not interested in any volume calculations. Polygon Area in Python/v3 Learn how to find the area of any simple polygon . Is it a Polygon? Answers (3) Sean de Wolski on 3 Dec 2013. Polygon is a closed figure with a given number of sides. So the area of this polygon-- there's kind of two parts of this. Click OK and it will automatically calculate the area for each polygon. You can calculate the area of a polygon by adding the areas of the trapezoids defined by the polygon’s edges dropped to the X-axis. Area of a quadrilateral. A = 1/2 ⋅ apothem ⋅ perimeter of polygon. We may notice, that during the calculations areas of are positive. number of sides n: n＝3,4,5,6.... circumradius r: side length a . Area of a polygon with given n ordered vertices in C++. This online calculator calculates the area of a polygon given lengths of polygon sides and diagonals, which split the polygon into non-overlapping triangles. So let's start with the area first. Area—Evaluates the area of the entire feature or its individual parts. Area is the two dimensional space inside the boundary of a flat object. Like to determine the area that wasn ’ t subtracted ( grey ) is the total distance covered a! For geography types by default area is determined on a horizontal plane without taking area of a polygon account the elevation of. Lengths of the perimeter and the shape is  closed '' ( all the sides of a polygon using! In ArcMap from the center of the irregular polygon. the x and coordinates! 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# Solving an equation involving complex conjugates

I have the following question and cannot seem to overcome how to contend with equations using $$z$$ and $$\bar z$$ together. For example, the below problem:

Find the value of $$z \in \Bbb C$$ that verifies the equation: $$3z+i\bar z=4+i$$

For other operations that didn't include mixing $$z$$ and $$\bar z$$, I was able to manage by "isolating" $$z$$ on one side of the equation and finding the real and imaginary parts of the complex numbers (sorry if I'm not using the right terms, it's my first linear algebra course)

I tried with wolfram and it didn't really help.

PS: I'm new to this forum but if it's like other math forums where they send you to hell if you ask for "help with your homework", this "homework" I'm doing is on my own since my semester is over and I just wanted to explore other subjects in the book that weren't covered in class.

• Have you tried picking a basis for $\mathbb{C}$, writing $z$ as a generic vector in that space using that basis, then solving for the components of $z$? I mean, ..., this is a linear algebra problem; why not "do the linear algebra thing" to it? – Eric Towers Dec 28 '18 at 2:29

Hint:

Let $$z = x + iy$$, for $$x,y \in \mathbb{R}$$. Consequently, $$\bar{z} = x - iy$$.

Make these substitutions into your equation and isolate all of the $$x$$ and $$y$$ terms on one side, trying to make it "look" like a number in that form above (I really don't know how else to describe it, my example below will be more illustrative).

Equate the real and imaginary parts to get a system of equations in two variables ($$x,y$$) which you can solve get your solution.

Similar Exercise To Show What I Mean:

Let's solve for $$z$$ with

$$iz + 2\bar{z} = 1 + 2i$$

Then, making our substitutions...

\begin{align} iz + 2\bar{z} &= i(x + iy) + 2(x - iy) \\ &= ix + i^2 y + 2x - 2iy \\ &= ix - y + 2x - 2iy \\ &= (2x - y) + i(x - 2y) \\ \end{align}

Thus,

$$(2x - y) + i(x - 2y) = 1 + 2i$$

The real part of our left side is $$2x-y$$ and the imaginary part is $$x - 2y$$. On the right, the real and imaginary parts are $$1$$ and $$2$$ respectively.

Then, we get a system of equations by equating real and imaginary parts!

\begin{align} 2x - y &= 1\\ x - 2y &= 2\\ \end{align}

You can quickly show with basic algebra that $$y = -1, x = 0$$.

Our solution is a $$z$$ of the form $$z = x + iy$$. Thus, $$z = 0 + i(-1) = -i$$.

One Final Tidbit:

PS: I'm new to this forum but if it's like other math forums where they send you to hell if you ask for "help with your homework", this "homework" I'm doing is on my own since my semester is over and I just wanted to explore other subjects in the book that weren't covered in class.

This forum doesn't mind helping you with homework, so long as you show you make a reasonable effort or at least have a clear understanding of the material. However, the goal is also to help you learn, so people tend to prefer nudges in the right direction if the context allows it, as opposed to just handing you the solution. (Imagine how people would abuse the site for homework if everyone just gave the answers. Not good, and not what math is about, you get me?)

• Then, we get a system of equations by equating real and imaginary parts!” OH ! This is so cool, didn’t know this could be done, but it does make sense. That’s what I was missing, thank you! As for the way you answered my question without exactly giving me the answer, that’s really what I was trying to get, an explanation and a line of reasoning so that I could then achieve it on my own. I like the mentality on this forum a lot so far. Have a good day :) – Laura Salas Dec 29 '18 at 21:26

Another approach is to take the complex conjugate of your equation: $$3\overline z-iz=4-i.$$ You now have two equations for $$z$$ and $$\overline z$$. Now eliminate $$\overline z$$ from them and solve for $$z$$.

• !! Also works, thank you. This is a shorter way to do it. – Laura Salas Dec 29 '18 at 21:27

If $$z=a+bi$$ then $$\bar z=a-bi$$

So you are solving: $$3(a+bi)+i(a-bi)=4+i$$ $$\to (3a+b)+(a+3b)i=4+i$$ Hence solve the simultaneous equations:

$$3a+b=4$$ $$a+3b=1$$

Let $$a$$ and $$b$$ be the real and imaginary parts of $$z$$. The equation becomes $$(3a+3ib)+i(a-ib)=4+i$$

Equating real and imaginary parts you get $$3a+b= 4$$ and $$3b+a=1$$. Now you should be able to discover that $$a=\frac {11} 8$$ and $$b =-\frac 1 8$$, so $$z=\frac {11} 8-i\frac 1 8$$.

• It’s $3a + 3ib$ in the first bracket of your first equation. – Live Free or π Hard Dec 28 '18 at 0:45
• @LiveFreeorπHard That was a typo. I had used $3a+3ib$ in the next step. Thanks anyway. – Kavi Rama Murthy Dec 28 '18 at 5:25


A subspace is a vector space that is contained within another vector space. So every subspace is a vector space in its own right, but it is also defined relative to some other (larger) vector space. We will discover shortly that we are already familiar with a wide variety of subspaces from previous sections. Here's the definition.

Definition S (Subspace) Suppose that $V$ and $W$ are two vector spaces that have identical definitions of vector addition and scalar multiplication, and that $W$ is a subset of $V$, $W\subseteq V$. Then $W$ is a subspace of $V$.

Lets look at an example of a vector space inside another vector space.

Example SC3: A subspace of $\complex{3}$.

## Testing Subspaces

In Example SC3 we proceeded through all ten of the vector space properties before believing that a subset was a subspace. But six of the properties were easy to prove, and we can lean on some of the properties of the vector space (the superset) to make the other four easier. Here is a theorem that will make it easier to test if a subset is a vector space. A shortcut if there ever was one.

Theorem TSS (Testing Subsets for Subspaces) Suppose that $V$ is a vector space and $W$ is a subset of $V$, $W\subseteq V$. Endow $W$ with the same operations as $V$. Then $W$ is a subspace if and only if three conditions are met

1. $W$ is non-empty, $W\neq\emptyset$.
2. If $\vect{x}\in W$ and $\vect{y}\in W$, then $\vect{x}+\vect{y}\in W$.
3. If $\alpha\in\complex{\null}$ and $\vect{x}\in W$, then $\alpha\vect{x}\in W$.

So just three conditions, plus being a subset of a known vector space, gets us all ten properties. Fabulous! This theorem can be paraphrased by saying that a subspace is "a non-empty subset (of a vector space) that is closed under vector addition and scalar multiplication."

You might want to go back and rework Example SC3 in light of this result, perhaps seeing where we can now economize or where the work done in the example mirrored the proof and where it did not. We will press on and apply this theorem in a slightly more abstract setting.

Example SP4: A subspace of $P_4$.

Much of the power of Theorem TSS is that we can easily establish new vector spaces if we can locate them as subsets of other vector spaces, such as the ones presented in Subsection VS.EVS:Vector Spaces: Examples of Vector Spaces.

It can be as instructive to consider some subsets that are not subspaces. Since Theorem TSS is an equivalence (see technique E) we can be assured that a subset is not a subspace if it violates one of the three conditions, and in any example of interest this will not be the "non-empty" condition. However, since a subspace has to be a vector space in its own right, we can also search for a violation of any one of the ten defining properties in Definition VS or any inherent property of a vector space, such as those given by the basic theorems of Subsection VS.VSP:Vector Spaces: Vector Space Properties. Notice also that a violation need only be for a specific vector or pair of vectors.

Example NSC2Z: A non-subspace in $\complex{2}$, zero vector.

Example NSC2A: A non-subspace in $\complex{2}$, additive closure.

There are two examples of subspaces that are trivial. Suppose that $V$ is any vector space. Then $V$ is a subset of itself and is a vector space. By Definition S, $V$ qualifies as a subspace of itself. The set containing just the zero vector $Z=\set{\zerovector}$ is also a subspace as can be seen by applying Theorem TSS or by simple modifications of the techniques hinted at in Example VSS. Since these subspaces are so obvious (and therefore not too interesting) we will refer to them as being trivial.

Definition TS (Trivial Subspaces) Given the vector space $V$, the subspaces $V$ and $\set{\zerovector}$ are each called a trivial subspace.

We can also use Theorem TSS to prove more general statements about subspaces, as illustrated in the next theorem.

Theorem NSMS (Null Space of a Matrix is a Subspace) Suppose that $A$ is an $m\times n$ matrix. Then the null space of $A$, $\nsp{A}$, is a subspace of $\complex{n}$.

Here is an example where we can exercise Theorem NSMS.

Example RSNS: Recasting a subspace as a null space.

## The Span of a Set

The span of a set of column vectors got a heavy workout in Chapter V:Vectors and Chapter M:Matrices. The definition of the span depended only on being able to formulate linear combinations. In any of our more general vector spaces we always have a definition of vector addition and of scalar multiplication. So we can build linear combinations and manufacture spans. This subsection contains two definitions that are just mild variants of definitions we have seen earlier for column vectors. If you haven't already, compare them with Definition LCCV and Definition SSCV.

Definition LC (Linear Combination) Suppose that $V$ is a vector space. Given $n$ vectors $\vectorlist{u}{n}$ and $n$ scalars $\alpha_1,\,\alpha_2,\,\alpha_3,\,\ldots,\,\alpha_n$, their linear combination is the vector \begin{equation*} \lincombo{\alpha}{u}{n}. \end{equation*}

Example LCM: A linear combination of matrices.

When we realize that we can form linear combinations in any vector space, then it is natural to revisit our definition of the span of a set, since it is the set of all possible linear combinations of a set of vectors.

Definition SS (Span of a Set) Suppose that $V$ is a vector space. Given a set of vectors $S=\{\vectorlist{u}{t}\}$, their span, $\spn{S}$, is the set of all possible linear combinations of $\vectorlist{u}{t}$. Symbolically,

\begin{align*} \spn{S}&=\setparts{\lincombo{\alpha}{u}{t}}{\alpha_i\in\complex{\null},\,1\leq i\leq t}\\ &=\setparts{\sum_{i=1}^{t}\alpha_i\vect{u}_i}{\alpha_i\in\complex{\null},\,1\leq i\leq t} \end{align*}

Theorem SSS (Span of a Set is a Subspace) Suppose $V$ is a vector space. Given a set of vectors $S=\{\vectorlist{u}{t}\}\subseteq V$, their span, $\spn{S}$, is a subspace.

Example SSP: Span of a set of polynomials.

Let's again examine membership in a span.

Example SM32: A subspace of $M_{32}$.

Notice how Example SSP and Example SM32 contained questions about membership in a span, but these questions quickly became questions about solutions to a system of linear equations. This will be a common theme going forward.

## Subspace Constructions

Several of the subsets of vectors spaces that we worked with in Chapter M:Matrices are also subspaces --- they are closed under vector addition and scalar multiplication in $\complex{m}$.

Theorem CSMS (Column Space of a Matrix is a Subspace) Suppose that $A$ is an $m\times n$ matrix. Then $\csp{A}$ is a subspace of $\complex{m}$.

That was easy! Notice that we could have used this same approach to prove that the null space is a subspace, since Theorem SSNS provided a description of the null space of a matrix as the span of a set of vectors. However, I much prefer the current proof of Theorem NSMS. Speaking of easy, here is a very easy theorem that exposes another of our constructions as creating subspaces.

Theorem RSMS (Row Space of a Matrix is a Subspace) Suppose that $A$ is an $m\times n$ matrix. Then $\rsp{A}$ is a subspace of $\complex{n}$.

One more.

Theorem LNSMS (Left Null Space of a Matrix is a Subspace) Suppose that $A$ is an $m\times n$ matrix. Then $\lns{A}$ is a subspace of $\complex{m}$.

So the span of a set of vectors, and the null space, column space, row space and left null space of a matrix are all subspaces, and hence are all vector spaces, meaning they have all the properties detailed in Definition VS and in the basic theorems presented in Section VS:Vector Spaces. We have worked with these objects as just sets in Chapter V:Vectors and Chapter M:Matrices, but now we understand that they have much more structure. In particular, being closed under vector addition and scalar multiplication means a subspace is also closed under linear combinations.


# Does the set of convex combination of points in Cantor set contains a non empty open interval?

$$\mathcal{C}$$ denote the cantor middle third set.

$$\mathcal{C}_t=\{(1-t)x+ty : x, y\in \mathcal{C} \}$$

$$\mathcal{C}_0=\mathcal{C}_1=\mathcal{C}$$ and we can prove that that $$\mathcal{C}$$ contains no non empty open interval.

What can be said for other $$t\in [0, 1]$$? Does it contains a non empty open interval ?

Can you list some resources where I can find such type of problems?

• Do you mean $\mathcal{C}_t=\{(1-t)x+ty : x, y\in \mathcal{C} \}$, with $t\in[0,1]$? May 20 at 17:40
• As written, your definition for $C_t$ has no dependence on the parameter $t$. I think you want to remove the $t \in [0,1]$ from the set-builder notation.
– Joe
May 20 at 18:08

While not a complete characterization of all the $$C_t$$, we may easily see that $$C_t$$ can contain a non-empty open interval for some values of $$t$$. Set $$t := \frac{1}{2}$$. Then we may compute:

\begin{align} C_{1/2} & = \{\frac{1}{2}x + \frac{1}{2}y : x,y \in C\} \\ & = \frac{1}{2} \cdot \{x + y : x,y \in C\}\\ & = \frac{1}{2} (C + C) \end{align} It’s easy to see from the “points in $$[0,1]$$ with ternary expansions consisting of only $$0$$s and $$2$$s” definition $$C$$ that $$C + C = [0,2]$$.

Therefore $$C_{1/2} = [0,1]$$.

EDIT: I gave it a little more thought, and we can say quite a bit. Let $$C^n$$ denote the $$n$$’th stage of the middle thirds construction of $$C$$, so that $$C = \bigcap_n C^n$$. I know this is non-standard notation, but I don’t want it to be confusing with $$C_t$$.

For $$\alpha \in [0,1]$$, we may easy see that: $$C_{\alpha} = \bigcap_{n} [\alpha C^n + \beta C^n]$$ Where $$\beta = (1 - \alpha)$$. Set $$X^n := \alpha C^n + \beta C^n$$. What does $$X^n$$ look like as we vary $$\alpha$$?

When $$\alpha \in \{0,1\}$$, we get that $$X^n = C^n$$, and we recover that $$C_0 = C_1 = C$$.

When $$\alpha = \frac{1}{2}$$, we get that $$X^n = [0,1]$$, and we recover that $$C_{1/2} = [0,1]$$.

What happens for $$\alpha \in (0, \frac{1}{2})$$? Well, we’ll have that $$C^n \subsetneq X^n$$. But we’ll also have that $$X^{n+1}$$ splits every interval in $$X^n$$. Hence we’ll end up with $$C_\alpha$$ being totally disconnected. Further, I believe that the measure of $$C_t$$ will monotonically increase as $$t$$ moves from $$0$$ to $$\frac{1}{2}$$, and then start monotonically decreasing again.

EDIT EDIT: I no longer believe this last part because it contradicts the paper in the other answer.

Can you list some resources where I can find such type of problems?

Maybe this is of interest:

Pawłowicz, Marta. Linear combinations of the classic Cantor set. Tatra Mt. Math. Publ. 56 (2013), 47–60.

From Math Review:

In this paper, linear combinations of classic Cantor sets are studied. The problem goes back to a result by Hugo Steinhaus [in Selected papers, 205–207, PWN, Warsaw, 1985], who proved in 1917 that $$C+C=[0,2]$$, where $$C$$ is the classic Cantor set and $$C+C=\{c_1+c_2; c_1,c_2∈C\}$$. This result was extended and generalized by several authors during the last hundred years. The main result of the present paper is the topological classification of linear combinations of $$C$$, i.e., sets of the form $$aC+bC=\{ac_1+bc_2; c_1,c_2∈C\}$$ where $$a,b∈R$$ are fixed. It is shown that this problem can be reduced to characterization of $$C+mC$$, where $$m∈(0,1)$$. This is given by the following theorem.

Theorem 1. $$C+mC=\bigcup_{n=1}^{2^k}[l_k^{(n)} ,r_k^{(n)}+m],$$for all $$m∈(0,1)$$, where $$k$$ is such that $$m∈[\frac{1}{3^{k+1}},\frac{1}{3^k})$$, $$k∈N_0$$, where $$l_k^{(n)}$$ and $$r_k^{(n)}$$ are the left and right endpoints of the $$n$$-th component of the $$k$$-th iteration of the Cantor set.


# Lecture 018

## Surjection - horizontal line test at least once

Surjection(surjectivity): everything in the codomain gets hit by something

• Definition Let A and B be sets and $f: A \rightarrow B$ be a function. f is surjective (or onto) iff $Im_f(A) = B$.

• $(\forall b \in B)(\exists a \in A)(f(a) = b)$

• $f:A \rightarrowtail B$

• then |A|>=|B| (B got all mapped even though there may be a overlap)

## Injection - horizontal line test at most once

Definition: let A, and B be set and $f: A \to B$ be a function, we say that f is injective(1-to-1).

• $(\forall x, y \in A)(f(x) = f(y) \implies x=y)$

• informal notation: $f: A \hookrightarrow B$

• Proof: let x, y s.t. f(x)=f(y), show x=y. You can show f(x)!=f(y) to by pass case check for piece-wise function.

• then |A|<=|B| (one to one, but not all B gets mapped)

## Bijection (Both Injection and Surjection)

Definition: let A, and B be set and $f: A \to B$ be a function, we say that f is bijection iff f is both injection and surjection.

• Proof: in two parts or

## Function Composition

Definition: Let A, B, C be sets and $f:A\to B \land g:B\to C$ be functions. The function $k:A\to C$ are defined by $(\forall a \in A)(h(a) = g(f(a)))$ is called the composition of g and f, denoted $h=g \circ f$.

Theorem: Let A, B, C, D be sets and $f:A\to B \land g:B\to C, h:C\to D$ be functions. Then $f \circ (g \circ f) = (h \circ g) \circ f$

• proof: $(h \circ (g \circ f))(a) = h((g\circ f)(a)) = h(g(f(a))) = (h \circ g)(f(a)) = ((h \circ g) \circ f)(a)$

observe: if $f: A\to A$, then $id_A \circ f = f \circ id_A = f$

## Identity Function

$id_A: A \to A, a |-> a$ TODO what is this notation TODO what is identity on a function

## Inverse

Definition: Let A, B be sets and $f: A \to B$ and $g: B \to A$ be functions. g is the inverse of f ($g = f^{-1})$ iff $f \circ g = id_B \land g \circ f = id_A$

Theorem: Let $f: A \to B$ be a function. f is invertible iff f is a bijection.

• prove forward: f is invertible -> f is a bijection

• prove 1-to-1
• invertible $(\exists g: B \to A)(g \circ f = id_A \land f \circ g = id_B)$
• let $a_1, a_2 \in A \land f(a_1) = f(a_2)$
• $g(f(a_1)) = g(f(a_2))$ by g well defined
• $id_A(a_1) = id_A(a_2)$ by $g \circ f = id_A$
• $a_1 = a_2$
• prove on-to
• let $b \in B$. Consider $a = g(b) \in A$. Since f and g are inverse. $f(a) = f(g(b)) = Id_B(b) = b$. Then f is subjective.
• prove backward: assume 1-to-1, onto.

• onto: $(\exists a \in A)(f(a) = b)$ -> at least one f(a) = b, fix such a
• 1-to-1: $(\forall x \in A)(x \neq a \implies f(x) \neq f(a) = b)$ -> at most one f(a) = b
• define $g = \{ (b, a) \in B \times A | f(a) = b\}$
• so g is a well defined
• let $a = g(b)$, then $f(a) = b$, then $f(g(b)) = b$ then $f \circ g = id_B$
• let $b = f(a)$, then $g(b) = a$, then $g(f(a)) = a$ then $g \circ f = id_A$
• $g = f^{-1}$

Corollary: if f is invertible, then f^-1 is unique

## Prove Bijection by Proving Invertible

Claim: $f: \mathbb{R} / \{3\} \to \mathbb{R} / \{1\}$ is a bijection by $f(x) = \frac{x-2}{x-3}$

Scratch:

• solve $x = \frac{3y - 2}{y - 1}$

• so $x \neq 3 \land y \neq 1$

Proof:

• define $g(x) = \frac{3x - 2}{x - 1}$

• observe $(\forall x \in \mathbb{R} \ \{1\})(g(x) \in \mathbb{R})$ because $x \neq 1$

• observe $g(x) \neq 3$ because $g(x)=3 \iff \frac{3x-2}{x-1} = 3 \iff 3x-2=3x-3 \iff -2=-3$

• so g is well-defined

• then $g(f(x)) = \frac{3\frac{3-2}{x-3}-2}{\frac{x-2}{x-3}-1} = x$ so $g \circ f = id_{\mathbb{R} \ \{3\}}$ holds

• show the same thing for $f(g(x))$

• f is invertible -> f is a bijection

Table of Content


# Diophantine equation with three variables

The question is:

Nadir Airways offers three types of tickets on their Boston-New York flights. First-class tickets are \$140, second-class tickets are \$110, and stand-by tickets are \$78. If 69 passengers pay a total of$6548 for their tickets on a particular flight, how many of each type of ticket were sold?

Now I set up my equation as

$140x+110y+78z=6548$

But I'm confused how to go from here. I know I need to find the GCD in order to evaluate that the equation has a solution and then set up my formulas for $x=x_{0}+\frac{b}{d}(n)$ and $y=y_{0}-\frac{a}{d}(n)$

Ive solved Diophantine equations before but only in the form $ax+by=c$. How do I continue from here? I'm not interested in the solution, I can do that by myself, but I would like to know the process from solving these types of Diophantine equations.

• Also $x+y+z=69$. I hope that you can find it. Have good days – scarface Oct 8 '17 at 19:32
• @scarface thank you! I can't believe I missed that, I feel so embarrassed for not realizing that. – user482578 Oct 8 '17 at 19:38
• $(x,y,z)=(9,19,41)$ – Donald Splutterwit Oct 8 '17 at 19:50
• After considering the sum of the passenger you should get $$31 x+16 y=583$$ – Raffaele Oct 8 '17 at 19:58

$140x+110y+78z=6548$

and

$x + y + z = 69$

$\implies 78x + 78y + 78z = 69*78 = 5382$

$\implies 62x + 32y = 1166 \implies 31x + 16y = 69*78 = 583$

And we can quickly deduce that $x = 9, y = 19, z = 41$ (by simple inspection in my case - using that we only have integer values for $x,y,z$.

If the $\gcd$ of the ticket prices does not divide the total revenue, then you are correct that there will be no integer solution. However you are not immediately guaranteed a solution if the $\gcd$ does divide the revenue, because we are constrained to non-negative numbers of tickets. So we could potentially run into a Frobenius-coin-type failure.

Here the total number of tickets reduces this to a simple "two-coin" problem:

\begin{align} &&140x+110y+78z &= 6548\\ \text{divide by }\gcd(x,y,z)=2&& 70x+55y+39z &= 3274\\ &&x+y+z &= 69\\ \text{multiply by }39 && 39x+39y+39z &= 2691\\ \text{subtract eqns} && 31x+16y &= 583\\ \bmod 16 && 31x\equiv 15x \equiv -1x&\equiv 583\equiv 7\\ \bmod 16 && x&\equiv -7\equiv 9\\ \text{test }x=25 && 31\cdot25 &= 775>583 \\ \text{thus }x=9 && 31\cdot 9 +16y&= 583 \\ && y= (583-279)/16 &= 19\\ && z= 69-(19+9) &= 41\\ \end{align}

In the reduced equation $31x+16y = 583$, since $583>(31{-}1)\cdot (16{-}1)$ the coin problem issue could not apply - the total is big enough to guarantee a solution.


# Math Help - Calculating position on a circumference

1. ## Calculating position on a circumference

Hello all, I hope someone is able to help me get my head round this little problem.

If I have a circle that is centered at (200,200) and its radius is 150, how do I calculate the point at any given angle? For example, I know that at 90 degrees, the point on the circumference will be (350,200), because I can calculate that manually, but what about more arbitrary degrees like 92.5 or 108?

Any help would be appreciated!

Thanks!

2. Originally Posted by gryphon5
Hello all, I hope someone is able to help me get my head round this little problem.

If I have a circle that is centered at (200,200) and its radius is 150, how do I calculate the point at any given angle? For example, I know that at 90 degrees, the point on the circumference will be (350,200), because I can calculate that manually, but what about more arbitrary degrees like 92.5 or 108?

Any help would be appreciated!

Thanks!
If the angle is 90 degrees, the point is (350,200)?
Do your angles start at the upper axis, the "North axis", then go clockwise?
If your angles start from the normal "East axis" then go counterclockwise, then the point at 90 degrees should be (200,350).

Whatever way you have there, for arbitrary angles/degrees, you just get the components of the 150-radius that are parallel to your axes.

Let us say your angles start the usual East-axis, or positive x-axis, and then go counterclockwise.

If the angle is 90 degrees,
x = 200 +150cos(90deg) = 200 +0 = 200
y = 200 +150sin(90deg) = 200 +150 = 350
Hence, point (200,350).

If the angle is 30 degrees,
x = 200 +150cos(30deg) = 200 +129.9 = 329.9
y = 200 +150sin(30deg) = 200 +75 = 275
Hence, point (329.9,275).

If the angle is 108 degrees,
x = 200 +150cos(108deg) = 200 -46.35 = 153.65
y = 200 +150sin(108deg) = 200 +142.66 = 342.66
Hence, point (153.65,342.66).

Etc....

3. Thanks vry much for your fast response.

Sorry for not including all the information, I should have stated...

This problem is for a computer application to draw an image, and since computers treat (0,0) as the top left of the screen (with the angles going clockwise), this is what I have used.

How would this change the examples you gave?

Thanks again!

4. Originally Posted by gryphon5
Thanks vry much for your fast response.

Sorry for not including all the information, I should have stated...

This problem is for a computer application to draw an image, and since computers treat (0,0) as the top left of the screen (with the angles going clockwise), this is what I have used.

How would this change the examples you gave?

Thanks again!
I see.

Then here is the change.

Let us call the vertical axis as y-axis also. The horizontal axis as x-axis also. The angles start from the upper or positive y-axis, going clockwise.
Our coordinates are in the usual (x,y) ordered pair.

If the angle is 30 degrees,
y = 200 +150cos(30deg) = 200 +129.9 = 329.9
x = 200 +150sin(30deg) = 200 +75 = 275
Hence, point (275,329.9).

If the angle is 108 degrees,
y = 200 +150cos(108deg) = 200 -46.35 = 153.65
x = 200 +150sin(108deg) = 200 +142.66 = 342.66
Hence, point (342.66,342.66).

If the angle is 92.5 degrees,
y = 200 +150cos(92.5deg) =200 -6.54 = 193.46
x = 200 +150sin(92.5deg) = 200 +149.86 = 349.86
Hence, point (349.86,193.46).

If the angle is 328.4 degrees,
y = 200 +150cos(328.4deg) = 200 +127.76 = 327.76
x = 200 +150sin(328.4deg) = 200 -78.60 = 121.40
Hence, point (121.40,327.76).

In other words,
For the x-component of the radius, use sine.
For the y-component of the radius, use cosine.
It's the reverse if you're doing them in the usual manner where the angles start from the positive x-axis, going counterclockwise.

5. Hello, gryphon5!

If I have a circle that is centered at (200,200) and its radius is 150,
how do I calculate the point at any given angle?
Code:
    |         * * *
|     *           *  P
|   *               *
|  *           r  / |*
|               /   |
| *           / θ   | *
| *        O* - - - + *
| *       (h,k)     Q *
|
|  *                 *
|   *               *
|     *           *
|         * * *
|
- + - - - - - - - - - - - - - - -
|

Consider a circle with radius $r$ with center $O(h,k)$.

Point $P$ creates $\angle POQ$ with the horizontal.

In right triangle $PQO$, we have:
. . $\cos\theta = \frac{OQ}{r}\quad\Rightarrow\quad OQ = r\cos\theta$
. . $\sin\theta = \frac{PQ}{r}\quad\Rightarrow\quad PQ = r\sin\theta$

The $x$-coordinate of $P$ is: . $x \:=\:h + OQ\:=\:h + r\cos\theta$
The $y$-coordinate of $P$ is: . $y \:=\:k + PQ \:=\:k + r\sin\theta$

Therefore, point $P$ is at: . $\left(h + r\cos\theta,\:k + r\sin\theta\right)$

6. Wow, very helpful, thanks to you both that has made things much clearer

7. Sorry to re-open an old thread, but the markup seems to have gone weird, is it possible for someone to ressurect it so I can read the equations again?

Thanks,
gryphon

8. Hello!

I'll try to format all this without LaTeX . . .

Code:
    |         * * *
|     *           *  P
|   *               *
|  *           r  / |*
|               /   |
| *           / θ   | *
| *        O* - - - + *
| *       (h,k)     Q *
|
|  *                 *
|   *               *
|     *           *
|         * * *
|
- + - - - - - - - - - - - - - - -
|

Consider a circle with radius r with center O(h,k).

Point P creates /POQ with the horizontal.

In right triangle PQO, we have:
. . cosθ = OQ/r . . OQ = r·cosθ
. . sinθ = PQ/r . . . PQ = r·sinθ

The x-coordinate of P is: .x .= .h + OQ .= .h + r·cosθ
The y-coordinate of P is: .y .= .k + PQ . = .k + r·sinθ

Therefore, point P is at: .(h + r·cosθ, k + r·sinθ)


# Determining Linearly Dependent Vectors

I am learning about Linear dependent vector from here

But I am unable to grasp the following equation:

If no such scalars exist, then the vectors are to be linearly independent.

$$c_1\begin{bmatrix}x_{11}\\x_{21}\\\vdots\\x_{n1}\\ \end{bmatrix}+c_2\begin{bmatrix}x_{12}\\x_{22}\\\vdots\\x_{n2}\\ \end{bmatrix}+\cdots+c_n\begin{bmatrix}x_{1n}\\x_{2n}\\\vdots\\x_{nn}\\ \end{bmatrix}=\begin{bmatrix}0\\0\\\vdots\\0\\ \end{bmatrix}\\ \begin{bmatrix}x_{11}&x_{12}&\cdots&x_{1n}\\x_{21}&x_{22}&\cdots&x_{2n}\\ \vdots&\vdots&\ddots&\vdots\\x_{n1}&x_{n2}&\cdots&x_{nn}&\\ \end{bmatrix}\begin{bmatrix}c_1\\c_2\\\vdots\\c_n\end{bmatrix}=\begin{bmatrix}0\\0\\\vdots\\0\end{bmatrix}$$ In order for this matrix equation to have a nontrivial solution, the determinant must be $0$

How the first equation is reduced to the second one?

• What exactly are you asking? – user418131 Sep 18 '18 at 9:05
• How the first equation is reduced to the second one? – Cody Sep 18 '18 at 9:06
• Write a formula for the $i$-th entry of the vector above and below and you will see they are the same. – Michal Adamaszek Sep 18 '18 at 9:08
• By the use of matrix multiplication. – user418131 Sep 18 '18 at 9:08
• The first equation is actually equivalent to $n$ equations. Do you know how to interchange linear equations with an analogous matrix equation? – user418131 Sep 18 '18 at 9:10

$$\begin{bmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{bmatrix}\begin{bmatrix} c_{1} \\ c_{2}\end{bmatrix}=\begin{bmatrix} c_{1}x_{11}+c_2x_{12} \\ c_{1}x_{21}+c_2x_{22} \end{bmatrix}=c_1\begin{bmatrix} x_{11} \\ x_{21}\end{bmatrix}+ c_2\begin{bmatrix} x_{21} \\ x_{22}\end{bmatrix}$$

Notice that $c_1$ is only multiplied to entries in the first column and $c_2$ is only multiplied to the entries in the second column.

This results from the definition of scalar multiplication and addition of matrices: \begin{align} c_1\begin{bmatrix} x_{11}\\x_{21}\\\vdots\\x_{n1} \end{bmatrix}+c_2\begin{bmatrix} x_{12}\\x_{22}\\\vdots\\x_{n2} \end{bmatrix}+\dots +c_n\begin{bmatrix} x_{1n}\\x_{2n}\\\vdots\\x_{nn} \end{bmatrix} &= \begin{bmatrix} c_1x_{11}\\c_1x_{21}\\\vdots\\c_1x_{n1} \end{bmatrix}+\begin{bmatrix} c_2x_{12}\\c_2x_{22}\\\vdots\\c_2x_{n2} \end{bmatrix}+\dots +\begin{bmatrix} c_nx_{1n}\\c_nx_{2n}\\\vdots\\c_nx_{nn} \end{bmatrix}\\[1ex] &= \begin{bmatrix} c_1x_{11}+c_2x_{12}+\dots+c_nx_{1n}\\c_1x_{21}+c_2x_{22}+\dots+c_nx_{2n}\\\dots\dots\dots\dots\dots\dots\dots\dots\\c_1x_{n1}+c_2x_{n2}+\dots+c_nx_{nn} \end{bmatrix} \end{align}

• So we are essentially writing a set of linear equations in matrix form, right? – Cody Sep 18 '18 at 9:54
• It's exactly that. – Bernard Sep 18 '18 at 10:00

Recall that the product $A\vec c$ can be interpreted as the linear combination of the colums $\vec x_i$ of $A$ by the coordinates $c_i$ of $\vec c$

$$A\vec c =\sum c_i\vec x_i$$

Refer also to the related


# Plot Transfer Function Matlab

Matlab also o ers w a ys to turn a sequence of graphs in toamo vie, con. You need to use the tf (link) function to produce a system object from your transfer function, and the lsim (link) function to do the simulation. Yes, i have Control System Toolbox. Bode Plot Example of First-Order System using Matlab. 2 in Control Systems By Nagoor Kani. Let me add to that last comment. Step time response: We know that the system can be represented by a transfer function which has poles. After reading the MATLAB control systems topic, you will able to solve problems based on the control system in MATLAB, and you will also understand how to write transfer function, and how to find step response, impulse response of various transfer systems. What does the MATLAB function ''tf2ss'' do ? Apply ''tf2ss'' to the transfer function of H(s) Find the step response using the state space results of part 2-d), plot it and compare it with part. I get the transfer function using. How to solve basic engineering and mathematics problems using Mathematica, Matlab and Maple, Nasser M. As a result this article presents an alternative that requires more lines of code but offers the full formatting flexibility of the generic plot command. a sensor with 0. Unformatted text preview: 6. A simple trick I found online was to use step() and divide the TF by s, and it should simulate a ramp response, step(G/s). Transfer function G(s) with plot or data. PI(D) Algorithm in MATLAB •We can use the pid() function in MATLAB •We can define the PI(D) transfer function using the tf() function in MATLAB •We can also define and implement a discrete. This studio will focus on analyzing the time response of linear systems represented by transfer function models. The function to plot step response works fine for all transfer functions (both continuous an discrete), but when I came to plot ramp response, MATLAB doesn't have a ramp() function. We can see that the PID controller we designed works well in the face of uncertainty in estimated transfer function parameters. Yes, i have Control System Toolbox. I want the graph to start at 5 after it leaves the transfer function block in Simulink. The name of the file and of the function should be the same. A SISO continuous-time transfer function is expressed as the ratio:. Hence, x-axis in your plot will only signify the total number of data points in FF_mag_nw. Numerator or cell of numerators. Plot Bode asymptote from Transfer Function. This plotting script employs the function cal_avg. It is obtained by applying a Laplace transform to the differential equations describing system dynamics, assuming zero initial conditions. Transfer functions can be used to represent closed-loop as well as open-loop systems. The ﬁrst parameter is a row vector of the numerator coeﬃcients. This shows how to use Matlab to solve standard engineering problems which involves solving a standard second order ODE. t is the time, ranging from 0 seconds to 10 seconds and w is a pulsation of 1. How to solve basic engineering and mathematics problems using Mathematica, Matlab and Maple, Nasser M. Transfer function G(s) with plot or data. The optical transfer function is not only useful for the design of optical system, it is also valuable to characterize manufactured systems. Add these time functions to produce the output. Plot transfer function response. rlocus(sys) calculates and plots the root locus of the open-loop SISO model sys. If needed, you can then convert the identified state-s[ace model into a transfer function using tf. The transfer function was $$\frac{20000}{s+20000}$$. MATLAB: A Practical Introduction to Programming and Problem Solving, winner of TAA’s 2017 Textbook Excellence Award ("Texty"), guides the reader through both programming and built-in functions to easily exploit MATLAB's extensive capabilities for tackling engineering and scientific problems. Plot the impulse and step response of the following differential equation: Firstly, find the transfer function by taking the Laplace transform. Running this m-file in the Matlab command window should gives you the following plot. The sys (system) structure in MATLAB v5 is very powerful, and it allows you to form complicated systems by joining together simpler systems. When a single vector argument is passed to plot, the elements of the vector form the dependent data and the index of the elements form the dependent data. If sys is a multi-input, multi-output (MIMO) model, then bode produces an array of Bode plots, each plot showing the frequency response of one I/O pair. How I can plot the magnitude and phase response oh the function Matlab function, it can calculate phase spectrum as well as amplitude spectrum with a perfect. how find ramp response. By applying Cauchy’s principle of argument to the open-loopsystem transfer function, we will get information about stability of the closed-loopsystem transfer function and arrive at the Nyquist stability criterion (Nyquist, 1932). A = logsig(N,FP) takes N and optional function parameters,. Title: 3D Plot Transfer Function Author: J. ( iddata or idfrd) where I gona used tfest function to estimate d transfer function. Function Plotting in Matlab. Note that the system transfer function is a complex function. Question: 9. Plot pole-zero diagram for a given tran. The Matlab function freqz also uses this method when possible ( e. RLocusGui is a graphical user interface written in the Matlab® programming language. (c) Clicking on the pole at 3/2 + √ 15/2 we see that Matlab predicts overshoot of 9. Hello, i am trying to make a bode plot of the transfer function of a twin-t notch filter, that i am analyzing. H(s) is a complex function and 's' is a complex variable. it has an amplitude and a phase, and ejωt=cosωt+jsinωt. Creates a continuous-time transfer function with numerator and denomi-nator speciﬁed by num and den. Control System Toolbox™ software supports transfer functions that are continuous-time or discrete-time, and SISO or MIMO. The transfer function is T s =. So the problem is how to run a Simulink model. More and more MATLAB users are using automation servers as part of continuous integration workflows. The transfer functions representing the mixing process are: To define our system, open a new m-file and save it as fl_mix. This studio will focus on analyzing the time response of linear systems represented by transfer function models. time response of a second order system 7. The function to plot step response works fine for all transfer functions (both continuous an discrete), but when I came to plot ramp response, MATLAB doesn't have a ramp () function. Frequency response plots show the complex values of a transfer function as a function of frequency. Then, you can apply any signal to the block and it will give you the output. ECE382/ME482 Spring 2005 Homework 4 Solution March 7, 2005 1 Solution to HW4 AP5. Question: 9. The root locus of an (open-loop) transfer function is a plot of the locations (locus) of all possible closed-loop poles with some parameter, often a proportional gain , varied between 0 and. i want write a script to plot a graph for the transfer function [H(f)] for a band pass filter, |H(f)| against frequency and the phase of H(f) (degrees) against frequency, im very new to matlab so the syntax is not 100%, im getting confused because everything is auto formatted in matrix form. The transfer function of a certain fourth-order, low pass, inverse Chebyshev filter with 3 dB frequency at 9600 radians/second will be used in all examples. Bode Plot Definition H. using % a) standard plotting and complex number capabilities, % b) standard plotting and complex number capabilities for generating Bode plots, and % c) built in Bode plot function. Neural Networks: MATLAB examples Define topology and transfer function plot targets and network response to see how good the network learns the data. If you want a different type of plot, look under Edit:Plot Configurations. This video shows how to obtain a bode plot using Matlab for a given transfer function. For example, consider the transfer function. And could tfest gives the transfer function where the data is in decibel. This function has three poles, two of which are negative integers and one of which is zero. There is a program within Matlab called Simulink. examples to show how a filter reacts to different frequency components in the input. A simple trick I found online was to use step() and divide the TF by s and it should simulate a ramp response, step(G/s). We can define the function having a scalar number as an input. 528 and no lag compensator, the. Make sure to "turn off" the feedback loop by setting the value of the gain to equal zero. You can add a controller, and compute the closed-loop transfer function. Run the simulation • Set the simulation to run for 30 seconds: Simulation->Configuration Parameters. sys_p is an identified transfer function model. It seems to me that the standard way of plotting the frequency response of the filter is to use a Bode plot. Add these time functions to produce the output. To construct a Bode plot from a transfer function, we use the following command:. I get the transfer function using. Plot the frequency spectrum (i. This tutorial discusses some of the different ways that MATLAB and Simulink interact. 5 (R2007b)] [Book]. Lattice or lattice ladder to transfer function. A SISO continuous-time transfer function is expressed as the ratio:. H is just the way to call what is the 'transfer matrix' of my system. on the plot and as thoroughly as you can, on the similarities and diﬀerences, if any, to the low–gain system step response from questions 8–9. In engineering, a transfer function (also known as system function or network function) of an electronic or control system component is a mathematical function which theoretically models the device's output for each possible input. Using MATLAB, input the transfer function H(s) H(s) = [ 23+3 s^2+2s+1]^T/( s^2+0. Transfer functions calculate a layer’s output from its net input. The figure produced by the bode(sys) function can be copied and pasted into wordprocessors and other programs. The plot displays the magnitude (in dB) and phase (in degrees) of the system response as a function of frequency. As a general rule, matlab programs should avoid iterating over individual samples whenever possible. Use the standard deviation data to create a 3σ plot corresponding to the confidence region. We are going to develop a function that will return the voltage and corresponding time of the response. A popular option is Jenkins. Using Matlab to create Transfer functions and bode plots. Let me add to that last comment. The optical transfer function is defined as the Fourier transform of the impulse response of the optical system, also called the point spread function. H is just the way to call what is the 'transfer matrix' of my system. Several examples of the construction of Bode Plots are included in this file. This MATLAB function computes the inverse Fast Fourier Transform of the optical transfer function (OTF) and creates a point-spread function (PSF), centered at the origin. sys = tf(B,A); t = 1:length(u); y = lsim(sys,u,t); figure plot(y) I am sure the estimated transfer function is correct, since it resembles the original system so much. Every digital filter can be specified by its poles and zeros (together with a gain factor). The roots of a(s) are called poles of the system. The frequency response of a system, is just the transfer function, evaluated at. Frequency Response, Bode Plots, and Resonance 3. You can multiply transfer functions sys1=tf(num1,den1) and sys2 = tf(num2, den2) using sys3=sys1*sys2. So the problem is how to run a Simulink model. A simple trick I found online was to use step() and divide the TF by s and it should simulate a ramp response, step(G/s). The result shown in the command window is ‘tf = empty transfer function’. A Bode plot is a graphical representation of a linear, time-invariant system transfer function. Bode introduced a method to present the information of a polar plot of a transfer function GH(s), actually the frequency response GH (jω), as two plots with the angular frequency were at the common axis. transfer function Eq. Click on the transfer function in the table below to jump to that example. How do I plot the contour of a given Nyquist plot onto the *s-plane of a given transfer function on Matlab? *s-plane: is the complex plane on which Laplace transforms are graphed. orF the original control system with K = 1. , RCL circuit with voltage across the capacitor C) as the output) is where is an arbitrary gain factor. The transfer function for the time delay can not be directly represented in MATLAB. Another thing is MATLAB plots infinity as one. More and more MATLAB users are using automation servers as part of continuous integration workflows. Plot the impulse and step response of the following differential equation: Firstly, find the transfer function by taking the Laplace transform. Yes, i have Control System Toolbox. My simulink model contain a bunch of 1/z unit delays, sums and gains. Transfer Function Representations. Of course you can, and T is called time delay. transfer function based on your choices, and compare the rise time, overshoot and damped oscillation frequency of the response you get from MATLAB with the corresponding values that you expect from the theory. Numerator or cell of numerators. Two transfer functions are combined to create a plant model. form the complete transfer function with the lag compensator added in series to th original system; plot the new Bode plot and determine phase margin and observe that it is the required phase margin; Now to do this In Matlab let us take a question. A transfer function is a convenient way to represent a linear, time-invariant system in terms of its input-output relationship. Calculate poles and zeros from a given transfer function. It is obtained by applying a Laplace transform to the differential equations describing system dynamics, assuming zero initial conditions. Bode Plot Definition H. The above plot shows that. In addition to estimating continuous-time transfer functions, System Identification Toolbox lets you estimate continuous-time state-space models and process models (special, low-order. The figure below shows a unity-feedback architecture, but the procedure is identical for any open-loop transfer function , even if some elements of the. Enter transfer function in MATLAB. Bode Plot Example of First-Order System using Matlab. Example1: Let us plot the Bode Plot for each transfer function and in doing so we will see the added functionality that can be achieved from the Bode plot function in MATLAB. How to make a plot with logarithmic axes in MATLAB ®. Plot of the disturbance model, called noise spectrum. purelin is a neural transfer function. Functions operate on variables within their own workspace, which is also called the local workspace, separate from the workspace you access at the MATLAB command prompt which is called the base workspace. Creating Bode Plots. This is achieved using the MATLAB-Simulink API (application program interface) commands. The function for step response works fine for all transfer functions (both continuous an discrete), but when I came to ramp response, MATLAB doesn't have a ramp() function. For MIMO models, pzmap displays all system poles and transmission zeros on a single plot. 3 MAXIMUM POWER TRANSFER 4. rlocus(sys) calculates and plots the root locus of the open-loop SISO model sys. I have done the calculations manually using Euler's formula, but now the assignment is asking me to compare these plots with the plots using freqz in MATLAB. Try this in matlab: s = tf('s'); T = 1; G = 1/s; Gd = exp(-s*T)/s; bode(G,Gd) and it will yield the following Bode diagram:. State space to transfer function. Plot the frequency spectrum (i. I've been trying to practice using Matlab for circuit analysis and am trying to create a transfer function plot of a high pass filter where the gain is in volts/volts not in dB. This way you can easily see how the two functions are similar or different from each other. So basically I have a digital filter and I need to plot a transfer function of this filter. This function creates arrows that go out from the origin of the axes in a polar coordinate system. logsig is a transfer function. The default formatting of most MATLAB plots is good for analysis but less than ideal for dropping into Word and PowerPoint documents or even this website. However, this would execute much slower because the matlab language is interpreted, while built-in functions such as filter are pre-compiled C modules. Estimating Other Model Types. Once plotted, you will. A SISO continuous-time transfer function is expressed as the ratio:. The transfer function generalizes this notion to allow a broader class of input signals besides periodic ones. 1 1]); >> grid >> grid The grids are optional. on the plot and as thoroughly as you can, on the similarities and diﬀerences, if any, to the low–gain system step response from questions 8–9. Also how to plot points on the bode plots and how to find help in Matlab. The first two software packages are free alternatives to Matlab, and their use is encouraged. A = purelin(N,FP) takes N and optional function parameters,. You can use static gain transfer function model sys1 obtained above to cascade it with another transfer function model. The roots of a(s) are called poles of the system. % Transfer function: 2500(10 + jw). RLocusGui is a graphical user interface written in the Matlab® programming language. MATLAB's tfestimate will produce a numerical estimate of the magnitude and phase of a transfer function given an input signal, an output signal, and possibly other information. Question: 9. and your transfer function is : 𝑉 𝑉𝑖 = 2 1+ 2 Now use the coefficients in MATLAB to plot the frequency response of this analog filter. Transfer functions calculate a layer’s output from its net input. CheungSlide 12. Transfer Function in MATLAB: As noted previously that the transfer function represents the input and output of the system in terms of the complex frequency variable so that the transfer function can give the complete information about the frequency response of the system. Then as a function of ω, the radian frequency, you plot the real and Imaginary parts from ω=0 to ω=∞. The transfer function is T s =. For example, consider the transfer function. Matlab also o ers w a ys to turn a sequence of graphs in toamo vie, con. Transfer functions calculate a layer's output from its net input. Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. You should get something like the following: To input this into MATLAB, choose some values for the resistor, capacitor and inductor constants: % Values for constants. There is a function bodeplot in Matlab which for instance takes an argument calculated with tf, which in turn takes a numerator and denominator. 1s + 1)(s+1)(10s+1). By applying Cauchy’s principle of argument to the open-loopsystem transfer function, we will get information about stability of the closed-loopsystem transfer function and arrive at the Nyquist stability criterion (Nyquist, 1932). zeros and poles from transfer function 3. Note that Eq. Use the standard deviation data to create a 3σ plot corresponding to the confidence region. % Transfer function: 2500(10 + jw). The root locus of an (open-loop) transfer function is a plot of the locations (locus) of all possible closed-loop poles with some parameter, often a proportional gain , varied between 0 and. ramp response of a transfer function 6. I have never in my whole life heard of a 3D transfer function, it doesn't make sense. The above command will plot FF_mag_nw which is a one-dimensional matrix against its row number as its x co-ordinate. To learn more about the use of functions, go through the user guide. The function depends on real input parameters. A SISO continuous-time transfer function is expressed as the ratio:. % There are some sample functions below that can be copied and pasted into the % proper location. Zeros, Poles and Pole–Zero Map of a Transfer Function The command tf2zp is used to obtain the zeros z, poles p and gain k of the transfer function … - Selection from MATLAB® and Its Applications in Engineering: [Based on MATLAB 7. From MATLAB command window, we will call the function CreatePlant to create the transfer function mentioned in shown: sys=CreatePlant(1,[1000 300 30 1]); step(sys) b. To plot more than one transfer function use the following syntax: bode(sys1,sys2,…). What you have to do is evaluate the transfer function's Real and Imaginary parts. So the problem is how to run a Simulink model. This way you can easily see how the two functions are similar or different from each other. Bode plot diagram state space. How to invert a transfer function in simulink (matlab)? I am having disturbance at the out put of my plant. In order to draw Bode Plot, we need transfer function from which we deduce the equations for Magnitude and Phase. transfer function based on your choices, and compare the rise time, overshoot and damped oscillation frequency of the response you get from MATLAB with the corresponding values that you expect from the theory. %simulate the estimated transfer function. Functions operate on variables within their own workspace, which is also called the local workspace, separate from the workspace you access at the MATLAB command prompt which is called the base workspace. Once you have converted your differential equation to a transfer function you can then use the options within the simulink control space to input your transfer function. Hello, lets say I have an image then I adjusted the contrast by using histogram equalization (histeq) Is there a way to plot or get an image of the transfer function that this command uses ?. not both the magnitude and and the phase. Then, you can apply any signal to the block and it will give you the output. It then shows how to generate the Bode plots and the step response plots of the transfer function. MATLAB provides command for working with transforms, such as the Laplace and Fourier transforms. I think you are completely wrong: z does not represent a complex number, but the fact that your transfer function is a discrete one, rather than a continuous one (see the Z transform for more details). So basically I have a digital filter and I need to plot a transfer function of this filter. Singular values plot of a transfer function. We can define the function having a scalar number as an input. I will be solving the question number 6. I want to have an equivalent input disturbance for the same. I've been trying to practice using Matlab for circuit analysis and am trying to create a transfer function plot of a high pass filter where the gain is in volts/volts not in dB. The locus of the roots of the characteristic equation of. Compare it to this, you want to plot a sine wave: x = sin(w*t), I hope you can agree with me that you cannot plot such a function (including axes) unless I specifically say e. purelin is a neural transfer function. Hello, lets say I have an image then I adjusted the contrast by using histogram equalization (histeq) Is there a way to plot or get an image of the transfer function that this command uses ?. Transfer functions are a frequenc view the full answer. State space controlability and observability. 1 GRAPH FUNCTIONS 2. System Stability If a linear system is described by a transfer function H(s), the system is said to be stable if. To export the linearized system to the Workspace so you can use it with other design tools in Matlab, select File: Export. I have never in my whole life heard of a 3D transfer function, it doesn't make sense. Transfer functions calculate a layer’s output from its net input. How can I plot two functions in the same graph?. I wanted to know how I can go about plotting a simple bode magnitude transfer function in LaTeX. Use the Laplace transform to solve for the time response and MATLAB for calculation and plotting. orF the original control system with K = 1. Singular values plot of a transfer function. That found the transfer function for the RL series circuit with a 20K resistor and a Inductor with a value of 500mH. The transfer function of the LTI system is the ratio of Laplace transform of output to the Laplace transform of input of the system by assuming all the initial conditions are zero. Bode Plot with Magnitude on a dB Scale in MATLAB % Magnitude of a Transfer Function on a dB Plot % Save output figures in bitmap mode for best quality. Hello, i am trying to make a bode plot of the transfer function of a twin-t notch filter, that i am analyzing. Abbasi [ next ] [ prev ] [ prev-tail ] [ tail ] [ up ] 1. 11/12/18 9 Oblique Wing Concepts •High-speed benefitsof wing sweep without the heavy structure and complex mechanism required for symmetric sweep •Blohmund Voss,R. You can also do the evaluation for negative values of ω, remembering that in the complex domain the point at infinity is a single point. The transfer function of a position control. If sys has transfer function. In this video I will give you a very quick but needed description of how to plot Step Response of Transfer Function Using Matlab. it has an amplitude and a phase, and ejωt=cosωt+ jsinωt. MATLAB proved very capable at taking the Bode plot of a given transfer function using the online documentation. Transfer functions calculate a layer’s output from its net input. I want to have an equivalent input disturbance for the same. Use the Laplace transform to solve for the time response and MATLAB for calculation and plotting. Of course in interpreting the Bode plot of an unknown system, one is seeing the plot of the entire system, and one must pick out the components from the whole. Here we will learn how to write a Matlab code for creating a transfer function and then analyzing this transfer code for its reaction to several. Without defined values of R and C you won't get any transfer function. The s is jw. Enter transfer function in MATLAB. Transfer Function Matlab Example. This function can be applied to any of the following negative feedback loops by setting sys appropriately. Open loop system response Figure 3 Open loop system To plot the open loop response, perform the following steps: a. This is achieved using the MATLAB-Simulink API (application program interface) commands. it has an amplitude and a phase, and ejωt=cosωt+jsinωt. Plot the bode diagram for the transfer function and: Plot the Bode Diagrams Find the Gain and Phase margin For what value of K, the closed-loop system is unstable? Plot the Nyquist plot by using MATLAB Repeat part (3). Estimating Other Model Types. Examine the Root Locus Diagram of the following transfer function. H is just the way to call what is the 'transfer matrix' of my system. to create s as a variable and then use s in a line of code to make a transfer function. MATLAB Answers. Plot the frequency spectrum (i. For example, consider the transfer function. Frequency response is usually a complex valued function, so it can be written as , where is the magnitude response and is the phase response. This means that the characteristic equation of the closed loop transfer function has no zeros in the right half plane (the closed loop transfer function has no poles there). The step function is one of most useful functions in MATLAB for control design. This plot is the same as a Bode plot of the model response, but it shows the output power spectrum of the noise model instead. Enter transfer function in MATLAB. 9 s + 6538 It gives us the transfer function for everything except the capacitor and my question is how do I convert this transfer function into its equvalent R and L. You need to use the tf (link) function to produce a system object from your transfer function, and the lsim (link) function to do the simulation. To construct a Bode plot from a transfer function, we use the following command:. Introduction. PI(D) Algorithm in MATLAB •We can use the pid() function in MATLAB •We can define the PI(D) transfer function using the tf() function in MATLAB •We can also define and implement a discrete. The transfer function generalizes this notion to allow a broader class of input signals besides periodic ones. Transfer functions calculate a layer’s output from its net input. This function is a modified version of the nyquist command, and has all the same attributes as the original, with a few improvements. bode automatically determines frequencies to plot based on system dynamics. Hello, lets say I have an image then I adjusted the contrast by using histogram equalization (histeq) Is there a way to plot or get an image of the transfer function that this command uses ?. When you call this function, you can specify system order as a vector, say [1 10], and the function will then return a plot helping you choose the best order as shown here. All the signals are transfer functions on the block diagrams. Plot Step Response of Transfer Function Using Simulink on Matlab. Root Locus with Time Delays. The plot function can accept one, two, or more arguments and produces a plot of the data contained in the arguments. Running this m-file in the Matlab command window should gives you the following plot. % You must edit this file under "** THE EQUATION: **" and enter the function y(s). Plot Bode asymptote from Transfer Function. A simple trick I found online was to use step() and divide the TF by s and it should simulate a ramp response, step(G/s). The Matlab Code: % This function creates two bode plots (amplitude and phase) for a transfer function. We can see that the PID controller we designed works well in the face of uncertainty in estimated transfer function parameters. Hello, i am trying to make a bode plot of the transfer function of a twin-t notch filter, that i am analyzing. % There are some sample functions below that can be copied and pasted into the % proper location. h = subplot(m,n,p), or subplot(mnp) breaks the Figure window into an m-by-n matrix of small axes, selects the pth axes object for for the current plot, and returns the axis handle. The aperiodic pulse shown below: has a Fourier transform: X(jf)=4sinc(4πf) As shown in MATLAB Tutorial #2, we can plot the amplitude and phase spectrum of this signal. My First problem is actually inputting the transfer function into Matlab, It's the transfer function of a first order hold which is: [(1 + sT) / T] x [(1 - e^-sT) / s] x [(1 - e^-sT) / s]. frequency - y-axiith 20is is the 20•l f th it d f th t flog of the magnitude of the transfer function in dB and x-axis is ω - xx -axis isaxis is ωorfor f inlogscalein log scale dB log(f) or log(ω) EE40 Fall 2009 Prof. Transfer functions calculate a layer's output from its net input. In this example, we will draw two graphs with the same function, but in second time, we will reduce the value of increment. Hence, x-axis in your plot will only signify the total number of data points in FF_mag_nw. Such plots are known as pole-zero plots. magnitude of a step input. MATLAB plotting commands, you should become familiar with the following commands: • tf - This command is used to enter transfer functions. 5 (R2007b)] [Book]. This function creates arrows that go out from the origin of the axes in a polar coordinate system. A transfer function is represented by 'H(s)'. transfer function and impulse response are only used in LTI systems. Ask Question Can I just find the frequency gain and then use SVD in Octave/MATLAB to plot every dot ?. Any advance for the correct way to use ‘tfest’. Transfer Function Analysis and Design Tools. Abbasi [ next ] [ prev ] [ prev-tail ] [ tail ] [ up ] 1. Write out your answer for the H(s). This is achieved using the MATLAB-Simulink API (application program interface) commands. , RCL circuit with voltage across the capacitor C) as the output) is where is an arbitrary gain factor. State space controlability and observability. If you want a different type of plot, look under Edit:Plot Configurations. There are two bode plots, one plotting the magnitude (or gain) versus frequency (Bode Magnitude plot) and another plotting the phase versus frequency (Bode Phase plot). Transfer Function Representations.


# For any two sets, $A - B = B - A$ implies $A = B$

Is the following statement True or False:

For any two sets $A$ and $B$: If $A - B = B - A$ then $A = B$.

If it is true, prove it, otherwise provide a counterexample.

I am unable to come up with a counter example. I think the statement is true but how do I prove it?

• Suppose $x \in A - B$. Then $x \in B-A$. In particular $x \in B$, contradiction. So $A - B = \emptyset$ i.e. $A \subset B$. Same argument shows $B - A = \emptyset$ i.e. $B \subset A$. – hunter Jun 23 '17 at 6:55

If $A-B=B-A$ then for any $x\in A-B=B-A$ we $x\in A;x\in B; x\not \in A; x\not \in B$. That's a contradiction so $A-B=B-A$ is empty.

Thus there are no elements in $A$ that are not in $B$. In other words $A$ is a subset of $B$. Likewise there are no elements of $B$ that are in $A$. So $B$ is a subset of $A$.

So $A=B$.

If $A \setminus B = B \setminus A$, then

$A=A \setminus B \cup (A\cap B)= B \setminus A \cup (B \cap A) = B$.

Let’s use some Boolean algebra, in order to show a different point of view.

Let $C=A\cup B$; for a subset $X$ of $C$, denote $X^c=C\setminus X$; thus $$A\setminus B=A\cap B^c,\qquad B\setminus A=B\cap A^c=A^c\cap B$$ Then \begin{align} A&=A\cap C && \text{because $A\subseteq C$} \\ &=A\cap (B\cup B^c) && \text{because $C=B\cup B^c$} \\ &=(A\cap B)\cup(A\cap B^c) && \text{distributivity} \\ &=(A\cap B)\cup(A^c\cap B) && \text{hypothesis} \\ &=(A\cup A^c)\cap B && \text{distributivity} \\ &=C\cap B && \text{because $A\cup A^c=C$} \\ &=B && \text{because $B\subseteq C$} \end{align}

You also have \begin{align} A\cap B^c &=(A\cap B^c)\cap(B\cap A^c) && \text{hypothesis} \\ &=A\cap(B^c\cap(B\cap A^c)) && \text{associativity} \\ &=A\cap((B^c\cap B)\cap A^c) && \text{associativity} \\ &=A\cap(\emptyset\cap A^c) && \text{because $B\cap B^c=\emptyset$} \\ &=A\cap\emptyset \\ &=\emptyset \end{align}