Prove Vector Space Properties Using Vector Space Axioms
Problem 711
Using the axiom of a vector space, prove the following properties.
Let $V$ be a vector space over $\R$. Let $u, v, w\in V$.
(a) If $u+v=u+w$, then $v=w$.
(b) If $v+u=w+u$, then $v=w$.
(c) The zero vector $\mathbf{0}$ is unique.
(d) For each $v\in V$, the additive inverse $-v$ is unique.
(e) $0v=\mathbf{0}$ for every $v\in V$, where $0\in\R$ is the zero scalar.
(f) $a\mathbf{0}=\mathbf{0}$ for every scalar $a$.
(g) If $av=\mathbf{0}$, then $a=0$ or $v=\mathbf{0}$.
(h) $(-1)v=-v$.
The first two properties are called the cancellation law.
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