# General Vector Spaces

## General Vector Spaces

Definition

A set $V$ is said to be a vector space over a scalar field $K$ if

(1) an addition operation “$+$” is defined between any two elements of $V$, and
(2) a scalar multiplication operation is defined between any element of $K$ and any element in $V$.

Moreover, the following properties must hold for all $u,v,w\in V$ and $a,b\in K$:

Closure Properties

(c1) $u+v\in V$.
(c2) $av\in V$.

(a1) $u+v=v+u$.
(a2) $u+(v+w)=(u+v)+w$.
(a3) There is an element $\mathbf{0}\in V$ such that $\mathbf{0}+v=v$ for all $v\in V$.
(a4) Given an element $v\in V$, there is an element $-v\in V$ such that $v+(-v)=\mathbf{0}$.

Properties of Scalar Multiplication
(m1) $a(bv)=(ab)v$.
(m2) $a(u+v)=au+av$.
(m3) $(a+b)v=av+bv$.
(m4) $1v=v$ for all $v\in V$.

The element $\mathbf{0} \in V$ is called the zero vector, and for any $v\in V$, the element $-v\in V$ is called the additive inverse of $v$.

=solution

### Problems

1. 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.