Given linear transformation T R2 rightarrow R3 Tx y x y xy

Given linear transformation T: R^2 rightarrow R^3, T(x, y) = (-x, y, x+y) B = {(1,1), (1,-1)} and B\' = {(1,1,0), (-1,1,0), (0,0,1)} Find transition matrix Q, from S_3 to B\'

Solution

we will have it as follow-;

Write as a combination of <1, 0, 0>, <0, 1, 1>, and <1, 1, 0>. That is, = a<1, 0, 0>+ b<0, 1, 1>+ c<1, 1, 0>= . So we have a+ c= x, b+ c= y, and b= z. Then c= y- z and a= x- y+ z.

That is, = (x-y+ z)<1, 0, 0>+ z<0, 1, 1>+ (y- z)<1, 1, 0>.

So T= (x- y+ z)T<1, 0, 0>+ zT<0, 1, 1>+ (y- z)T<1, 1, 0>. is the transition