Okay, I did most of the parts in this question, just wanted you guys to check if it's the correct answer and teach me on the parts that I don't know how to do! Thanks a lot guys.

Question :

Let S denote the standard basis for and B = { , } be another basis.

(a) write down the change of basis matrix P B->S, from the basis B to the basis S.

P B->S =

(b) Hence find the change of basis matrix P S->B, from the basis S to the basis B.

P S-> B = inverse of (P B->S)

= inverse of

= 1/2

(c) Find if x = (4,-1)

I don't know how to do this one, can you guys help me out?

(d) Show that defined by :

Let x = [ x1, x2], y = y1, y2]

T(x+y)= T(x1 + y1, x2 + y2) = (-x1 + 2y1, 3y1)

= ( -x1, 0.x2) + (2y1, 3y2)

= T(x1,x2) + T(y1,y2)

T(ax) = T(ax1, ax2)

= (-ax1, a.0)

= a (-x1, 0)

= aT(x)

(e) Find the matrix representation of T with respect to the standard basis

=

(f) Use your answers from above to find the matrix representation of T with respect to the basis B,

=

(g) Find

where

=

= [tex][T]_B[x]_B =