Linear Algebra Please show work I dont understand this fully

Linear Algebra

Please show work. I dont understand this fully.

3. Let T: R^3 - > R^2 be the linear transformation with matrix representation A = [ ] relative to ordered bases B, B\' where B = {(1, -1,0), (1,1,0), (0,0, 1)}. Find the coordinate vector [T(1, -3, 3)]B\'. (Hint: you do not need to know what vectors are in B\'.)

Solution

B\'=B*A

=[1 -1 0 [1 2

1 1 0 3 1

0 0 1 ] * 4 2 ]

= [1-3 2-1

1+3 2+1

4 2 ]

=[-2 1

4 3

4 2]

[T(1,-1,3)]B\' = T*B = [ 1 -3 3 ] * [ -2 1 = [-2-12+12 1-9+6] = [ -2 -2 ]

4 3

4 2]