[SOLVED] Math 231 — hw 6

1. Prove or disprove if this is a vector space using theorem 1.34 from the textbook:
{(x1, x2, x3) | x1x2x3 = 0, xi ∈ R}.
2. Construct an example of a vector space W with two subspaces, W1, W2 where W1 +
W2 ̸= W.
3. Let V = R
3
, and define two subspaces:
• V1 = {(x, y, 0) | x, y ∈ R}
• V2 = {(0, y, z) | y, z ∈ R}
Prove that V1 + V2 forms a subspace of V .
4. Prove that V1 + V2 = V in the previous problem.
5. Let V = R
3
, and define two subspaces:
• V1 = {(x, y, 0) | x + y = 0, x, y ∈ R}
• V2 = {(0, y, z) | y + z = 0, y, z ∈ R}
Prove or provide a counter example to the statement: V1 + V2 = V.