/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q2.7-10Q Consider the following geometric... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Q2.7-10Q (page 93)

Consider the following geometric 2D transformations: D, a dilation (in which x-coordinates and y-coordinates are scaled by the same factor); R, a rotation; and T a translation. Does D commute with R? That is, is \(D\left( {R\left( {\bf{x}} \right)} \right) = R\left( {D\left( {\bf{x}} \right)} \right)\)for all \({\bf{x}}\) in \({\mathbb{R}^{\bf{2}}}\)? Does D commute with T? Does R commute with T?

Short Answer

Expert verified

D commutes with R, and matrices D and R do not commute with T.

Step by step solution

01

Write the transformation matrix for dilation, rotation, and translation

Let the transformation matrices in homogenous coordinates for dilation, rotation and translation be

\(D = \left[ {\begin{array}{*{20}{c}}s&0&0\\0&s&0\\0&0&1\end{array}} \right]\), \(R = \left[ {\begin{array}{*{20}{c}}{\cos \phi }&{ - \sin \phi }&0\\{\sin \phi }&{\cos \phi }&0\\0&0&1\end{array}} \right]\), and \(T = \left[ {\begin{array}{*{20}{c}}1&0&h\\0&1&k\\0&0&1\end{array}} \right]\).

02

Compute the product of transformation matrices

Calculate the value of \(DR\).

\(DR = \left[ {\begin{array}{*{20}{c}}{s\cos \phi }&{ - s\sin \phi }&0\\{s\sin \phi }&{s\cos \phi }&0\\0&0&1\end{array}} \right]\)

Calculate the value of \(RD\).

\[RD = \left[ {\begin{array}{*{20}{c}}{s\cos \phi }&{ - s\sin \phi }&0\\{s\sin \phi }&{s\cos \phi }&0\\0&0&1\end{array}} \right]\]

Calculate the value of \(DT\).

\(DT = \left[ {\begin{array}{*{20}{c}}s&0&{sh}\\0&s&{sk}\\0&0&1\end{array}} \right]\)

Calculate the value of \(TD\).

\(TD = \left[ {\begin{array}{*{20}{c}}s&0&h\\0&s&k\\0&0&1\end{array}} \right]\)

Calculate the value of \(RT\).

\(RT = \left[ {\begin{array}{*{20}{c}}{\cos \phi }&{ - \sin \phi }&{h\cos \phi - k\sin \phi }\\{\sin \phi }&{\cos \phi }&{h\sin \phi + k\cos \phi }\\0&0&1\end{array}} \right]\)

Calculate the value of \(TR\).

\(TR = \left[ {\begin{array}{*{20}{c}}{\cos \varphi }&{ - \sin \varphi }&h\\{\sin \varphi }&{\cos \varphi }&k\\0&0&1\end{array}} \right]\)

03

Check the commutation of transformation matrices

For the transformation matrices, \(DR = RD\).

So, D commutes with R.

\(DT \ne TD\)

So, D does not commute with T.

And

\(RT \ne TR\)

So, R does not commute with T.

Hence, D commutes with R, and matrices D and R do not commute with T.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Suppose Ais a \(3 \times n\) matrix whose columns span \({\mathbb{R}^3}\). Explain how to construct an \(n \times 3\) matrix Dsuch that \(AD = {I_3}\).

A useful way to test new ideas in matrix algebra, or to make conjectures, is to make calculations with matrices selected at random. Checking a property for a few matrices does not prove that the property holds in general, but it makes the property more believable. Also, if the property is actually false, you may discover this when you make a few calculations.

38. Use at least three pairs of random \(4 \times 4\) matrices Aand Bto test the equalities \({\left( {A + B} \right)^T} = {A^T} + {B^T}\) and \({\left( {AB} \right)^T} = {A^T}{B^T}\). (See Exercise 37.) Report your conclusions. (Note:Most matrix programs use \(A'\) for \({A^{\bf{T}}}\).

In exercise 11 and 12, mark each statement True or False.Justify each answer.

a. If \(A = \left[ {\begin{array}{*{20}{c}}{{A_{\bf{1}}}}&{{A_{\bf{2}}}}\end{array}} \right]\) and \(B = \left[ {\begin{array}{*{20}{c}}{{B_{\bf{1}}}}&{{B_{\bf{2}}}}\end{array}} \right]\), with \({A_{\bf{1}}}\) and \({A_{\bf{2}}}\) the same sizes as \({B_{\bf{1}}}\) and \({B_{\bf{2}}}\), respectively then \(A + B = \left[ {\begin{array}{*{20}{c}}{{A_1} + {B_1}}&{{A_{\bf{2}}} + {B_{\bf{2}}}}\end{array}} \right]\).

b. If \(A = \left[ {\begin{array}{*{20}{c}}{{A_{{\bf{11}}}}}&{{A_{{\bf{12}}}}}\\{{A_{{\bf{21}}}}}&{{A_{{\bf{22}}}}}\end{array}} \right]\) and \(B = \left[ {\begin{array}{*{20}{c}}{{B_1}}\\{{B_{\bf{2}}}}\end{array}} \right]\), then the partitions of \(A\) and \(B\) are comfortable for block multiplication.

[M] Suppose memory or size restrictions prevent your matrix program from working with matrices having more than 32 rows and 32 columns, and suppose some project involves \(50 \times 50\) matrices A and B. Describe the commands or operations of your program that accomplish the following tasks.

a. Compute \(A + B\)

b. Compute \(AB\)

c. Solve \(Ax = b\) for some vector b in \({\mathbb{R}^{50}}\), assuming that \(A\) can be partitioned into a \(2 \times 2\) block matrix \(\left[ {{A_{ij}}} \right]\), with \({A_{11}}\) an invertible \(20 \times 20\) matrix, \({A_{22}}\) an invertible \(30 \times 30\) matrix, and \({A_{12}}\) a zero matrix. [Hint: Describe appropriate smaller systems to solve, without using any matrix inverse.]

Use partitioned matrices to prove by induction that for \(n = 2,3,...\), the \(n \times n\) matrices \(A\) shown below is invertible and \(B\) is its inverse.

\[A = \left[ {\begin{array}{*{20}{c}}1&0&0& \cdots &0\\1&1&0&{}&0\\1&1&1&{}&0\\ \vdots &{}&{}& \ddots &{}\\1&1&1& \ldots &1\end{array}} \right]\]

\[B = \left[ {\begin{array}{*{20}{c}}1&0&0& \cdots &0\\{ - 1}&1&0&{}&0\\0&{ - 1}&1&{}&0\\ \vdots &{}& \ddots & \ddots &{}\\0&{}& \ldots &{ - 1}&1\end{array}} \right]\]

For the induction step, assume A and Bare \(\left( {k + 1} \right) \times \left( {k + 1} \right)\) matrices, and partition Aand B in a form similar to that displayed in Exercises 23.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.