This module will define what a vector space is and provide useful examples to the reader.
Introduction
Vector space
A vector space
$S$ is a collection of "vectors" such that (1) if
${f}_{1}\in S\implies \alpha {f}_{1}\in S$ for all scalars
$\alpha $ (where
$\alpha \in \mathbb{R}$ ,
$\alpha \in \mathbb{C}$ , or some other field) and (2) if
${f}_{1}\in S$ ,
${f}_{2}\in S$ , then
$({f}_{1}+{f}_{2})\in S$
To define an vector space, we need
A set of things called "vectors" (
$X$ )
A set of things called "scalars" that form a field (
$A$ )
A vector addition operation (
$$ )
A scalar multiplication operation (
$*$ )
The operations need to have all the properties of givenbelow. Closure is usually the most important to show.
Vector spaces
If the scalars
$\alpha $ are real,
$S$ is called a
real vector
space .
If the scalars
$\alpha $ are complex,
$S$ is called a
complex
vector space .
If the "vectors" in
$S$ are functions
of a continuous variable, we sometimes call
$S$ a
linear function
space
Properties
We define a set
$V$ to be a vector space if
$x+y=y+x$ for each
$x$ and
$y$ in
$V$
$x+(y+z)()=(x+y)()+z$ for each
$x$ ,
$y$ , and
$z$ in
$V$
There is a unique "zero vector" such that
$x+0=x$ for each
$x$ in
$V$ (0 is the field additive identity)
For each
$x$ in
$V$ there is a unique vector
$-x$ such that
$x+-x=0$
$1x=x$ (1 is the field multiplicative identity)
$({c}_{1}{c}_{2})x={c}_{1}({c}_{2}x)$ for each
$x$ in
$V$ and
${c}_{1}$ and
${c}_{2}$ in
$\u2102$
$c(x+y)=cx+cy$ for each
$x$ and
$y$ in
$V$ and
$c$ in
$\u2102$
$({c}_{1}+{c}_{2})x={c}_{1}x+{c}_{2}x$ for each
$x$ in
$V$ and
${c}_{1}$ and
${c}_{2}$ in
$\u2102$
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