Question

Suppose V and W are two vector spaces. We can make the set V × W = {(α, β)|α ∈ V,β ∈ W} into a vector space as follows:

(α1,β1)+(α2,β2)=(α1 + α2,β1 + β2)

c(α1,β1)=(cα1, cβ1)

You can assume the axioms of a vector space hold for V × W

(A) If V and W are finite dimensional, what is the dimension of V × W? Prove your answer.

Now suppose W1 and W2 are two subspaces of V . We can define a linear map

Φ : W1 × W2 → V by Φ(α, β) = α + β

(B): Show R(Φ) = W1 + W2

(C): Show dim(N(Φ)) = dim(W1 ∩ W2) Hint: Define T : W1 ∩ W2 → V × W by T(α)=(α,−α)

Answer 1