Let T𝑇 and S𝑆 be linear transformations defined by T(x,y)=(2x−3y,−y)𝑇(𝑥,𝑦)=(2𝑥−3𝑦,−𝑦) and S(x,y)=(x−y,x)𝑆(𝑥,𝑦)=(𝑥−𝑦,𝑥). What is the matrix corresponding to the linear transformation T∘S𝑇∘𝑆?

To find the matrix corresponding to the linear transformation T∘S, we first need to find the result of the composition of the transformations T and S.

The transformation S(x,y) is given by (x-y, x). We substitute this into the transformation T, which gives us T(S(x,y)) = T(x-y, x) = (2(x-y) - 3x, -x).

Simplify this to get T(S(x,y)) = (-x - 2y, -x).

The matrix corresponding to this transformation is obtained by considering the images of the basis vectors under the transformation. The basis vectors in R^2 are (1,0) and (0,1).

T(S(1,0)) = T(1,1) = (-1, -1) and T(S(0,1)) = T(-1,0) = (2, 0).

So, the matrix corresponding to the transformation T∘S is given by the columns (-1, -1) and (2, 0). That is,

[-1 2]
[-1 0]