Answer to Question #277974 in Linear Algebra for Aashish

Question

For any two subspace W1,W2 of R^3 of dimension 2, W1+ W2 is a direct sum . True or false with full explanation

Answer 1

Its true.

0 = 0 + 0 ∈ W1 + W2 and so it is nonempty.

Let α, β ∈ F. Let w1 + w2, w0 1 + w0 2 ∈ W1 + W2 where w1, w0 1 ∈ W1, w2, w0 2 ∈ W2. Since W1

and W2 are subspaces, αw1 + βw0 1 ∈ W1 and αw2 + βw0 2 ∈ W2. Hence α(w1 + w2) +

β(w0 1 + w0 2 ) = (αw1 + βw0 1 ) + (αw2 + βw0 2 ) ∈ W1 + W2