1. The problem statement, all variables and given/known data Prove that if a∈F (where F represents ℝ or ℂ), v∈V (where V is a vector space) and av = 0, then a= 0 or v = 0. 2. Relevant equations The axioms for a vector space may be relevant. 3. The attempt at a solution Case 1 (v = 0): Suppose that a∈F, v∈V, and av = 0. Also, let u∈F. Then av + au = au (added au to both sides) So a(v + u) = au (distributive property) And v + u = u (multiplied both sides by 1/a) Therefore v = 0. Case 2 (a = 0): Suppose that a∈F, v∈V, and av = 0. I'm stuck here. The book proves this statement slightly different. For Case 1, the author simply assumes a ≠ 0 and he may therefore multiply both sides by 1/a to get v = 0. For Case 2, the author more or less says av = 0 when a = 0 just because, without any kind of justification. On a side note, I just finished a proofs book and began self-studying linear algebra for mathematicians. The above problem is from Linear Algebra Done Right (by Axler), which I recently read is targeted at graduate students. As an undergraduate, I'm having a hell of a time so far with this book. If anyone can recommend a more intermediate linear algebra book (undergraduate level, with proofs) I'd surely appreciate it. Thanks in advance.