[SOLVED] Math 231 — hw 8

1. Let V be a vector space and 0 ∈ V the additive identity. Prove that 0 + 0 = 0. Then
prove that 0 + . . . + 0 = 0 for any finite number of sums.
2. Let V = R
3 and consider the subspaces:
W1 = {(x, y, 0) | x, y ∈ R}, W2 = {(0, 0, z) | z ∈ R}.
Prove that V = W1 ⊕ W2 using the last theorem from class.
3. Let V = R
3
. Consider the subspace U = {(x, y, 0) | x + y = 0}. Find a space W such
that V = U ⊕ W.