The following definitions are used in Exercises 23–30.
Definition. If S1 and S2 are nonempty subsets of a vector space V, then the sum of S1 and S2, denoted S1+S2, is the set {x+y : x ∈ S1 and y ∈ S2}.
Definition. A vector space V is called the direct sum of W1 and W2 if W1 and W2 are subspaces of V such that W1 ∩ W2 = {0} and W1 +W2 = V. We denote that V is the direct sum of W1 and W2 by writing V = W1 ⊕ W2.
Let F be a field that is not of characteristic 2. Define W 1 = {A ∈ M n×n (F): A ij = 0 whenever i ≤ j} and W 2 to be the set of all symmetric n × n matrices with entries from F. Both W 1 and W 2 are subspaces of M n×n (F). Prove that M n×n (F) = W 1 ⊕ W 2 . Compare this exercise with Exercise 28.
28. A matrix M is called skew-symmetric if M t = −M. Clearly, a skewsymmetric matrix is square. Let F be a field. Prove that the set W 1 of all skew-symmetric n × n matrices with entries from F is a subspace of M n×n (F). Now assume that F is not of characteristic 2 (see Appendix C), and let W 2 be the subspace of M n×n (F) consisting of all symmetric n × n matrices. Prove that M n×n (F) = W 1 ⊕ W 2 .
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