Solved: Let (V, +V , V ) and (W, +W , W ) be vector spaces and define V W = {(v, w) : v
Chapter 4, Problem 36(choose chapter or problem)
Let (V, +V , V ) and (W, +W , W ) be vector spaces and define V W = {(v, w) : v V and w W}. Prove that (a) V W is a vector space, under componentwise operations. (b) if S = {(v, 0) : v V} and S = {(0, w) : w W}, then S and S are subspaces of V W. (c) if dim[V] = n and dim[W] = m, then dim[V W] = m + n. [Hint: Write a basis for V W in terms of bases for V and W.]
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