Question

Let T1 and T2 be linear transformations given by T1 x1 x2 = 3x1 + 6x2 −2x1 + 7x2 T2 x1 x2 = −2x1 + 8x2 6x2 . Find the matrix A such that the following is true. (a) T1(T2(x)) = Ax (b) T2(T1(x)) = Ax (c) T1(T1(x)) = Ax (d) T2(T2(x)) = Ax

Answer 1

To find the matrix A, apply the given linear transformations to the vector x. The matrix representations for (a) T1(T2(x)), (b) T2(T1(x)), (c) T1(T1(x)), and (d) T2(T2(x)) are [-6, 48], [-4, 56], [-2, 42], and [-2, 112] respectively.

Step-by-step explanation:

To find the matrix A that satisfies the given conditions, we need to apply the transformations T1 and T2 to the vector x. Let's start with (a) T1(T2(x)):

T2(x) = -2x1 + 8x2 + 6x2 = -2x1 + 14x2

T1(T2(x)) = T1(-2x1 + 14x2) = 3(-2x1 + 14x2) + 6x2 = -6x1 + 42x2 + 6x2 = -6x1 + 48x2

Therefore, the matrix representation for (a) T1(T2(x)) is [ -6, 48 ].

We can follow the same steps to find the matrix representations for (b) T2(T1(x)), (c) T1(T1(x)), and (d) T2(T2(x)).

(b) T2(T1(x)) matrix representation is [ -4, 56 ]

(c) T1(T1(x)) matrix representation is [ -2, 42 ]

(d) T2(T2(x)) matrix representation is [ -2, 112 ]