Suppose T and U are linear transformations from R to Rn such that T(Ux) -x for all x in R", Is it true that UTx)-x for all x in Rn? 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. A. No, it is not true. ABT is the standard matrix for T(u(x)). By hypothesis, T(U(x))- x is the identity mapping and so ABL. However, this does not imply that BAT I, where BAT is the standard matrix for U(T(x). so U(T(x)) is not necessarily the identity matrix.
B. Yes, it is true. AB is the standard matrix of the mapping x+T(U(x)) due to how matrix multiplication is defined. By hypothesis, this mapping is the identity mapping, so AB Since both A and B are square and AB-, them Invertible Matrix Theorem states that both A and B invertible, and B A-. Thus, BA I. This means that the mapping xU(T(x) is the identity mapping. Therefore, U(T(x))- x for all x in R.
C. Yes, it is true. AB is the standard matrix for T(U(x). By hypothesis, T(U(x)-x is the trivial mapping and so AB = 0. This implies that either A or B is the zero matrix, and so BA 0. This implies that U(T(x)) is also the trivial mapping.
D. No, it is not true. AB is the standard matrix for T(Ux) By hypothesis, T(Ux)- x is the identity mapping and so AB-I. However, matrix multiplication is not commutative, so BA is not necessarily equal to I. Since BA is the standard matrix for U(Tx)). U(T(x)) is not necessarily the identity matrix.