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 V denote the vector space consisting of all upper triangular n × n matrices (as defined in Exercise 12), and let W1 denote the subspace of V consisting of all diagonal matrices. Show that V = W 1 ⊕ W 2 , where W 2 = {A ∈ V: A ij = 0 whenever i ≥ j}.
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