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 ∈ Rn×n|AT = −A}
that is the set of all skew symmetric matrices, is a subspace. Here, we have denoted by AT the transpose of A, that is the matrix {aTij} = AT defined by aTij = aji.
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 TA : Rn → Rn denote the linear operator defined by
TAx = A · x,
for every x ∈ Rn. Prove that R(TA) 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 Tk to denote the composition of T with itself k times.
(i) Show that for every k ∈ N one has R(Tk+1) ⊂ R(Tk).
(ii) Show that there exists a positive integer m such that for all k ≥ m one has R(Tk) = R(Tk+1).
5. Let A and B be two square, n × n matrices. Prove that if AB = 0 (as matrix products), then
R(TA) + R(TB) ≤ n,
where we have denoted by TA and TB the linear operators associated to the matrices A and B respectively.
