[Solved] MA502 Homework 4

1. Consider the set of all n n real matrices. This set is has a vector space structure, as we have seen in class. Prove that

S = {A Rⁿⁿ|A^T = A}

that is the set of all skew symmetric matrices, is a subspace. Here, we have denoted by A^T the transpose of A, that is the matrix {a^T_ij} = A^T defined by a^T_ij = a_ji.

2. Consider T : P3 P2 defined by differentiation, i.e., by T(p) = p⁰ P2 for p P3. Find the range and the Null space for T.
3. Let A be a n n matrix with real coefficients and let T_A : Rⁿ Rⁿ denote the linear operator defined by

T_Ax = A x,

for every x Rⁿ. Prove that R(T_A) is equal to the span of the columns of A.

4. Let T : Rⁿ Rⁿ be a linear operator and for every k N set T^k to denote the composition of T with itself k
   • Show that for every k N one has R(T^k+1) R(T^k).
   • Show that there exists a positive integer m such that for all k m one has R(T^k) = R(T^k+1).
5. Let A and B be two square, n n Prove that if AB = 0 (as matrix products), then

R(T_A) + R(T_B) n,

where we have denoted by T_A and T_B the linear operators associated to the matrices A and B respectively.