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.
In M m×n (F) define W 1 = {A ∈ M m×n (F): A ij = 0 whenever i > j} and W 2 = {A ∈ M m×n (F): A ij = 0 whenever i ≤ j}. (W 1 is the set of all upper triangular matrices defined in Exercise 12.) Show that M m×n (F) = W 1 ⊕ W 2 .
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