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Chapter 2: Q8E (page 57)

Question 8: Find the inverse of linear transformation

y1=x1+7x2y2=3x1+20x2

Short Answer

Expert verified

Answer:

Inverse of the transformation isT-1(x1,x2)=(-20x1+7x2,3x1-x2) .

Step by step solution

01

Step1: System of equations

We have given a linear transformation with

y1=x1+7x2y2=3x1+20x2

02

Linear Transformation

Let T be linear transformation from R2鈫�R2.

Then, according to given equations.

Tx1x2=x1+7x23x1+20x2

03

Matrix for linear transformation

The matrix representation of T by 2x2 matrix.

.A=17320

Inverse of matrix is calculated as

First of all we will find the determinant of matrix.

A=20-21=-1adjA=20-7-31

Inverse of matrix= .A-1=-2073-1

Hence, inverse of the transformation will be

T-1(x1,x2)=(-20x1+7x2,3x1-x2)

Hence, inverse of the transformation isdata-custom-editor="chemistry" T-1(x1,x2)=(-20x1+7x2,3x1-x2)

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Most popular questions from this chapter

Consider the regular tetrahedron sketched below, whose center is at the

origin.

Let T from 鈩�3to 鈩�3be the rotation about the axis through the points 0 and P2that transforms P1 into P3. Find the images of the four corners of the tetrahedron under this transformation.

P0鈫�rP1鈫�P3P2鈫�P3鈫�

Let L from 鈩�3to 鈩�3be the reflection about the plane through the points 0 , p0andp3. Find the images of the four corners of the tetrahedron under this transformation.

P0鈫�LP1鈫�P2鈫�P3鈫�

Describe the transformations in parts (a) through (c) geometrically.

a.T-1b.L-1c.T2=T鈭�T鈥�(thecompositeofTwithitself)

d. Find the images of the four corners under the transformations T鈭�LandL鈭�T. Are the two transformations the same?

P0鈫�T鈭�LP0鈫�T鈭�LP1鈫�P1鈫�P2鈫�P2鈫�P3鈫�P3鈫�

e. Find the images of the four corners under the transformation L鈭�T鈭�L. Describe this transformation geometrically

Question: For two invertible n 脳 n matrices Aand B, determine which of the formulas stated in Exercise 67 through 75 are necessarily true.

(A鈭�B)(A+B)=A2鈭�B2

In Exercises 55 through 64, find all matricesX that satisfy the given matrix equation. 64.

123456X=I2

Consider a matrix Aof the form A=[abb鈭�a] , where a2+b2=1. Find two nonzero perpendicular vectors蠀鈫�andlocalid="1664256213921" w鈫�such thatA蠀鈫�=蠀鈫�and Aw鈫�=鈭�w鈫�(write the entries of 蠀鈫�and w鈫�in terms of aand b). Conclude that localid="1664256257917" T(x鈫�)=A(x鈫�)represents the reflection about the line Lspanned by蠀鈫�.

Consider an n 脳 p matrix A, a p 脳 m matrix B, and a scalar k. Show that (k A)B = A (k B) = k(AB)
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