15.1 Vector spaces

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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

1. $x+y=y+x$ for each $x$ and $y$ in $V$
2. $x+(y+z)()=(x+y)()+z$ for each $x$ , $y$ , and $z$ in $V$
3. There is a unique "zero vector" such that $x+0=x$ for each $x$ in $V$ (0 is the field additive identity)
4. For each $x$ in $V$ there is a unique vector $-x$ such that $x+-x=0$
5. $1x=x$ (1 is the field multiplicative identity)
6. $\left({c}_{1}{c}_{2}\right)x={c}_{1}\left({c}_{2}x\right)$ for each $x$ in $V$ and ${c}_{1}$ and ${c}_{2}$ in $ℂ$
7. $c(x+y)=cx+cy$ for each $x$ and $y$ in $V$ and $c$ in $ℂ$
8. $({c}_{1}+{c}_{2})x={c}_{1}x+{c}_{2}x$ for each $x$ in $V$ and ${c}_{1}$ and ${c}_{2}$ in $ℂ$

Examples

• $\mathbb{R}^{n}=\mathrm{real vector space}$
• $\mathbb{C}^{n}=\mathrm{complex vector space}$
• ${L}^{1}(\mathbb{R})=\{f(t)\colon \int_{()} \,d t\}$ f t f t is a vector space
• is a vector space
• ${L}^{2}(\mathbb{R})=\{f(t)\colon \int_{()} \,d t\}$ f t 2 f t finite energy signals is a vector space
• ${\ell }^{1}(\mathbb{Z})$ , ${\ell }^{2}(\mathbb{Z})$ , ${\ell }^{\infty }(\mathbb{Z})$ are vector spaces
• The collection of functions piecewise constant between the integers is a vector space

• ${ℝ}_{+}^{2}=\{\left(\begin{array}{c}{x}_{0}\\ {x}_{1}\end{array}\right)\colon ({x}_{0}> 0)\land ({x}_{1}> 0)\}$ is not a vector space. $\left(\begin{array}{c}1\\ 1\end{array}\right)\in {ℝ}_{+}^{2}$ , but $\forall \alpha , \alpha < 0\colon \alpha \left(\begin{array}{c}1\\ 1\end{array}\right)\notin {ℝ}_{+}^{2}$
• $D=\{\forall z, \left|z\right|\le 1\colon z\in \mathbb{C}\}$ is not a vector space. $({z}_{1}=1)\in D$ , $({z}_{2}=i)\in D$ , but $({z}_{1}+{z}_{2})\notin D$ , $\left|{z}_{1}+{z}_{2}\right|=\sqrt{2}> 1$

Vector spaces can be collections of functions, collections of sequences, as well as collections of traditionalvectors ( i.e. finite lists of numbers)

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in general
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