Suppose T and U are linear transformations from to such that T(Ux)x for all x in . Is it true that U(Tx)x for all x in ? Why or why not? Let A be the standard matrix for the linear transformation T and B be the standard matrix for the linear transformation U. Choose the correct answer below. A. Yes, it is true. AB is the standard matrix of the mapping due to how matrix multiplication is defined. By hypothesis, this mapping is the identity mapping, so ABI. Since both A and B are square and ABI, the Invertible Matrix Theorem states that both A and B invertible, and B. Thus, BAI. This means that the mapping is the identity mapping. Therefore, U(T(x))x for all x in . B. No, it is not true. AB is the standard matrix for . By hypothesis, x is the identity mapping and so ABI. However, matrix multiplication is not commutative, so BA is not necessarily equal to I. Since BA is the standard matrix for , is not necessarily the identity matrix. C. Yes, it is true. AB is the standard matrix for . By hypothesis, x is the trivial mapping and so AB0. This implies that either A or B is the zero matrix, and so BA0. This implies that is also the trivial mapping. D. No, it is not true. AB is the standard matrix for . By hypothesis, x is the identity mapping and so ABI. However, this does not imply that BAI, where BA is the standard matrix for . So is not necessarily the identity matrix.