Vector Space

Def) Vector Space

A vector space is a set, usually denoted as $V$ that is paired with a field F, such that they hold the following properties : 1)

Vector Space Axioms:

Vector Addition(Notation : +) : $\begin{array}{rl}V\times V& \to V\\ (v_1,v_2)& \mapsto v_1+v_2\end{array}$ Axioms :

1. $v_1+(v_2+v_3)=(v_1+v_2)+v_3$
2. $0+v=v$
3. (v) + (-v) = 0

Scalar Multiplication(Notation : Juxtaposition) : $\begin{array}{rl}F\times V& \to V\\ (a,v)& \mapsto av\end{array}$ Axioms :

1. $1v=v$
2. $a(bv)=(a\cdot b)v$

Compatibility (aka: Distributive Property)

1. $a(v_1+v_2)=a{v}_1+a{v}_2$
2. $(a+b)v=av+bv$

Elementary Consequence

Prove the following statements using the axioms above : $0v=0$ $a0=0$

exampls :

We can define $R^n$ and $C^n$ as vector spaces over $R$ and $C$ respectively

Example : Define $R$ as a vector space over R

vector addition: $2+3=5\in R$ Scalar Multiplication : $(2)3=6\in R$

$R^3$ vector addition:

$(1,2,3)+(1,2,3)=(2,4,6)$

scalar multiplication : $5(1,2,3)=(5\cdot 1,5\cdot 2,5\cdot 3)=(5,10,15)$

Example : $Fun(S,F)$ forms a vector space over F

Let