4. (4 points each) In each of the following, you are given two vector spaces V and W, with bases

By and Bw, respectively, and a linear transformation T: V → W. Determine the matrix of

T with respect to By and Bw. Determine the rank and nullity of T.

(a) V = R³, W = R², By =

(b) V = R₂[X], W

-(0)·()·())-²- - ()-(-))-

Bw

=

3

defined by

(c) V = W = M2x2 (R), By

fined by

-0-0)

T

=

= R¹, By = (1, X, X², X³), Bw

N

=

T(a+bX+cX²) =

2a

a

T (05) - (²

Т

C d

-(0000)--

=

b+c

a

b

C

(a+b+c)

Bw

- (( ) ( ) ( ) ( . )). T I. de

=

is

T is defined by

b+c

6+)

2d