T R2 rightarrow R2 rotates points about the origin through

T : R^2 rightarrow R^2 rotates points (about the origin) through -pi /4 radians (clockwise).

Solution

Since the final vector is of size 2X1 , hence the matrix size of T will be 2X2 matrix

The e1 vector will be [1 0]

Let the matrix T be 2X2 matrix as given below

by multiplying with e1 we get

a(1) + b(0) = 1/sqrt(2)

c(1) + d(0) = -1/sqrt(2)

hence we get a = 1/sqrt(2) and c = -1/sqrt(2), since it must be equal to 3pi/4 shift

Hence multiplying with e2 vector i.e. [0,1] vector will yield

by multiplying with e1 we get

a(0) + b(1) = 1/sqrt(2)

c(0) + d(1) = 1/sqrt(2)

Hence the final matrix will be equal to

a	b
c	d