Examples of a Vector Space

The following are examples of Vector Spaces

Zero vector space §

The set $V=\{\vec{0}\}$ is the simplest vector space if addition and multiplication are defined as normal.

In this space, it doesn’t matter what field we fix it over, as the axioms will always hold true.

#todo Show that all the axioms are satisfied, as $\vec{0}$ is the additive inverse of itself.

Vector space of ${\mathbb{R}}^n$ §

The real vector space is defined with both the scalars being $\mathbb{F}=\mathbb{R}$. Let ${\mathbb{R}}^n$ be the set of all real vectors with $n$ dimensions. E.g. ${\mathbb{R}}^2$ is the set of all two dimensional real Vectors fixed over the field $\mathbb{R}$.

Vector space of Matrices §

Matrices can also be defined as a vector space.

Let $M_{m,n}(\mathbb{F})$ be the set of all matrices of size $m\times n$ which has all entries being elements of the set $\mathbb{F}$ For example, $M_{2,2}(\mathbb{R})$ is the set of all $2\times 2$ matrices which have real entries (entries in $\mathbb{R}$)

$M_{m,n}$ is a vector space when addition is defined to be the sum of entries, and multiplication is defined to be Matrix Multiplication:

Vector space of Polynomials §

Let ${\mathcal{P}}_n(\mathbb{F})$ be the set of all polynomials with degree $\le n$ (where $n\in \mathbb{Z},n\ge 0$) having coefficients being elements of set $\mathbb{F}$.

$${\mathcal{P}}_n(\mathbb{F})=\{a_0+a_{1}x+a_2{x}^2+a_3{x}^3+\dots a_n{x}^n|a_0,a_1,a_2\dots a_n\in \mathbb{F}\}$$

${\mathcal{P}}_n$ is a vector space when fixed over the

Warning $\le n$ is DIFFERENT FROM $=n$

Sometimes, you’ll see ${\mathcal{P}}_n$ being used to define the set of polynomials with degree equal to $n$. This is not a vector space, because you can take $\vec{u}=x^n+1$ and $\vec{v}=x^n$, which both are valid $n$-degree polynomials. Subtracting (adding the inverse of) $\vec{v}$ from $\vec{u}$ yields: $\vec{u}+{\vec{v}}^{\prime }=x^n+1+(-x^n)=1\notin {\mathcal{P}}_n$

Vector space of Functions §

Let $\mathcal{F}(S_1,S_2)$ be the set of all functions that map from $S_1$ to $S_2$, where $S_1$ is a non-empty set. $$\vec{f}: S_{1} \to S_{2} \iff \vec{f} \in \mathcal{F(S_{1},S_{2})}

Let $\vec{f}, \vec{g} \in \mathcal{F}(S,\mathbb{R})$ i.e. $\vec{f}$ and $\vec{g}$ are two functions that map elements from $S$ to the real numbers. Fix $\mathcal{F}$ over the [Field](field.md) $\mathbb{R}$ and define vector addition and scalar multiplication as follows: #TODO

(\vec{f} + \vec{g})(x) =

### Vector space of $\mathbb{F}_{2}^n$ The set $\mathbb{F}^n_{2}$ is all the vectors of dimensions $n$ which has components in the finite field of two elements, $\mathbb{F}_{2}$. That is,

\vec{v} \in \mathbb{F}{2}^n \iff \vec{v} = (a{1}, a_{2}, \dots a_{n})| a_{1}, a_{2},\dots a_{n}\in \mathbb{F}_{2}

#TODO This vector space has useful applications in computer science, such as in [Hamming Codes](Hamming%20Codes) ### Vector space of $\mathbb{C}^n$ Also called the complex vector space, this is a vector space with field fixed over the complex numbers, i.e. $\mathbb{F} = \mathbb{C}$

\vec{v} \in \mathbb{C}^n \iff \vec{v} = (a_{1},a_{2},a_3\dots a_{n})| a_{1},a_{2},a_{3},\dots a_{n} \in \mathbb{C}$$
The normal Complex Numbers belong to the vector space ${\mathbb{C}}^1=\mathbb{C}$

All the vector spaces above can be expressed as a complex vector space:

Real Vector Space	Complex Vector Space
${\mathbb{R}}^n$	${\mathbb{C}}^n$
$M_{m,n}(\mathbb{R})$	$M_{m,n}(\mathbb{C})$
${\mathcal{P}}_n(\mathbb{R})$	${\mathcal{P}}_n(\mathbb{C})$
$\mathcal{F}(S,\mathbb{R})$	$\mathcal{F}(S,\mathbb{C})$
* Assuming that both vector spaces are fixed over their respective fields i.e. Real vector spaces are fixed over the $\mathbb{R}$ field and complex vector spaces are fixed over the $\mathbb{C}$ field.