Show that V=W W for the given vector space V and the given subspaces W and...

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Show that V=W₁ W₂ for the given vector space V and the given subspaces W₁ and W₂. (a) V = R³, W₁ = {(x₁, x2, x3): x₁ + x₂ + x3 = 0}, W₂ = {(0, 0, x3): x3 = R}. (b) V = M3×3 (C), W₁ = {A: Ajj = 0 for i > j}, W₂ = {A: Aj = 0 for i ≤ j}. (c) V = P₂ (R), W₁ = {f(x): f(x) = f(-x)}, W₂ = {f(x): f(-x) = -f(x)}. Show that V=W₁ W₂ for the given vector space V and the given subspaces W₁ and W₂. (a) V = R³, W₁ = {(x₁, x2, x3): x₁ + x₂ + x3 = 0}, W₂ = {(0, 0, x3): x3 = R}. (b) V = M3×3 (C), W₁ = {A: Ajj = 0 for i > j}, W₂ = {A: Aj = 0 for i ≤ j}. (c) V = P₂ (R), W₁ = {f(x): f(x) = f(-x)}, W₂ = {f(x): f(-x) = -f(x)}.

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