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Chapter 2: Q34E (page 54)

Consider the transformation Tfrom R2to role="math" localid="1659714471562" R2that rotates any vector x鈫�through a given angle胃in the counterclockwise direction. Compare this with Exercise 33. You are told that Tis linear. Find the matrix of Tin terms of胃.

Short Answer

Expert verified

Matrix for T will becos胃-sin胃sin胃cos胃

Step by step solution

01

Step by step Explanation: Step1: Linear Transformation

A transformation from Rm鈫�Rnis said to be linear if the following condition holds:

  1. Identity of Rmshould be mapped to identity of Rn.
  2. T(a+b)=T(a)+T(b)For all a,b鈭�Rm.
  3. T(ca)=cT(a)Where c is any scalar and a鈭�Rm.
02

Transformation

Given T is linear transformation fromR2鈫�R2.

Let be any point ofR2.

Then we can write the transformation.

T(rcos伪,rsin伪)=(rcos(伪+胃),rsin(伪+胃))=(rcos伪cos胃-rsin伪sin胃,rsin胃cos伪+rcos胃sin伪)=cos胃-sin胃sin胃cos胃rcos伪rsin伪T尾=cos胃-sin胃sin胃cos胃

Hence, the matrix for T will be[cos胃-sin胃sin胃cos胃].

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Most popular questions from this chapter

Consider two n x nmatrices A and B whose entries are positive or zero. Suppose that all entries of A are less than or equal to 鈥�s鈥�, and all column sums of B are less than or equal to 鈥�r鈥� (the column sum of a matrix is the sum of all the entries in its column). Show that all entries of the matrix AB are less than or equal to 鈥�sr鈥�.

If A is any transition matrix and B is any positive transition matrix, then AB must be a positive transition matrix.

If A is a 3脳4matrix and B is a4脳5, then AB will be a5脳3matrix.

TRUE OR FALSE?

There exists a real number k such that the matrix [k-23-3k-2]fails to be invertible.

TRUE OR FALSE?

If a matrixA=(abcd) represents the orthogonal projection onto a line L , then the equationa2+b2+c2+d2=1must hold.

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