linear transformation matrix?

Question

T:R2-->R2 is defined as T:(x,y)= (x+y,x-y)..
Using the bases B1=B2= {(1,-1), (-3,2)} find the transformation matrix.

please elaborate..the answer is listed as

-6 17
-2 6

Answer 1

The last link was the most informative.
The function is as follows:
(x' y')T = [(1 1)T (1 -1)T] (x y)T

If you multiply the basis vectors by this matrix you get:
(0 2)T for the (1 -1)T vector and
(-1 -5)T for the (-3 2)T vector.

(0, 2) = c1 * (1, -1) + c2 * (-3, 2)
so:
0 = c1 - 3 * c2
2 = - c1 + 2 * c2
c1 = -6
c2 = -2

likewise:
(-1, -5) = c3 * (1, -1) + c4 * (-3, 2)
c3 = 17 and c4 = 6

M = [(-6 -2)T (17 6)T]

Answer 2

The matrix A is 4x6. The transformation makes use of matrix multiplication to coach a vector x with 4 components right into a vector with 6. The area is R^4 and the codomain R^6. As for b. i do no longer recognize what it capacity by a picture so i won't be able to help out.

Answer 3

you need to do the following:
T( (1,-1))= (0, 2) = -6 (1,-1) - 2 (-3,2) so the first column of your matrix is:
-6
-2

now
T ( (-3,2) ) = ( -1, -5) = 17 (1,-1) + 6 (-3,2), so the second column of your matrix is:
17
6

when you put them together you get:
-6 17
-2 6
'

Answer 4

1 -3 I 1 0
-1 2 I 0 1

1 -3 I 1 0
0 -1 I 1 1

1 0 I -2 -3
0 -1 I 1 1

1 0 I -2 -3
0 1 I -1 -1
T^-1=(w1,w2)=(-2w1-3w2,-w1-w2)
the things on top are matrices and its inverse...hope its helpful:)