## 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**.