15.1 Vector spaces

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 $(f_1+f_2)\in S$

• 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 ( $*$ )

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. $(c_1{c}_2)x=c_1(c_{2}x)$ 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
• $L^{\infty }(\mathbb{R})=\{f(t)\colon f\left(t\right)\mathrm{ is bounded}\}$ 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
• $L^2(\left[0 , T\right]())=\mathrm{finite energy functions on interval\; [0,T]}$
• ${\ell }^1(\mathbb{Z})$ , ${\ell }^2(\mathbb{Z})$ , ${\ell }^{\infty }(\mathbb{Z})$ 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 $(z_1+z_2)\notin D$ , $\left|z_1+z_2\right|=\sqrt{2}> 1$