Description
1. Determine whether the following set is a vector space. Justify your answer using
theorem 1.34 from the textbook:
W = {(x, y, z) | x − (y + 1) + 2(z + 1) = 1, x, y, z ∈ R}.
2. Construct an example of a vector space W with two subspaces, W1, W2, where you
know W1 + W2 = W. Attempt to prove this.
3. Let V = R
4
, and define two subspaces:
• V1 = {(x, y, 0, 0) | x, y ∈ R}
• V2 = {(0, 0, z, w) | z, w ∈ R}
Prove that V1 + V2 forms a subspace of V .
4. Prove that V1 + V2 = V in the previous problem.
5. Let V = R
2
, and define two subspaces:
• V1 = {(w1, w2) | w1 + 2w2 = 0, w1, w2 ∈ R}
• V2 = {(v1, v2) | v1 + v2 = 0, v1, v2 ∈ R}
Prove or provide a counterexample to the statement: V1 + V2 = V .