# Review of linear algebra

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Vector spaces are the principal object of study in linear algebra. A vector space is always defined with respectto a field of scalars.

## Fields

A field is a set $F$ equipped with two operations, addition and mulitplication, and containing two special members 0 and 1( $0\neq 1$ ), such that for all $\{a, b, c\}\in F$

• $(a+b)\in F$
• $a+b=b+a$
• $\left(a+b\right)+c=a+\left(b+c\right)$
• $a+0=a$
• there exists $-a$ such that $a+-a=0$
• $ab\in F$
• $ab=ba$
• $abc=abc$
• $a1=a$
• there exists $a^{(-1)}$ such that $aa^{(-1)}=1$
• $a(b+c)=ab+ac$
More concisely
• $F$ is an abelian group under addition
• $F$ is an abelian group under multiplication

,,

## Vector spaces

Let $F$ be a field, and $V$ a set. We say $V$ is a vector space over $F$ if there exist two operations, defined for all $a\in F$ , $u\in V$ and $v\in V$ :

• vector addition: ( $u$ , $v$ ) $(u+v)\in V$
• scalar multiplication: ( $a$ , $v$ ) $av\in V$
and if there exists an element denoted $0\in V$ , such that the following hold for all $a\in F$ , $b\in F$ , and $u\in V$ , $v\in V$ , and $w\in V$
• $u+\left(v+w\right)=\left(u+v\right)+w$
• $u+v=v+u$
• $u+0=u$
• there exists $-u$ such that $u+-u=0$
• $a(u+v)=au+av$
• $(a+b)u=au+bu$
• $abu=abu$
• $1u=u$
More concisely,
• $V$ is an abelian group under plus
• Natural properties of scalar multiplication

## Examples

• $\mathbb{R}^{N}$ is a vector space over
• $\mathbb{C}^{N}$ is a vector space over
• $\mathbb{C}^{N}$ is a vector space over
• $\mathbb{R}^{N}$ is not a vector space over
The elements of $V$ are called vectors .

## Euclidean space

Throughout this course we will think of a signal as a vector $x=\left(\begin{array}{c}{x}_{1}\\ {x}_{2}\\ \\ {x}_{N}\end{array}\right)=\begin{pmatrix}{x}_{1} & {x}_{2} & & {x}_{N}\\ \end{pmatrix}^T$ The samples $\{{x}_{i}\}$ could be samples from a finite duration, continuous time signal, for example.

A signal will belong to one of two vector spaces:

## Real euclidean space

$x\in \mathbb{R}^{N}$ (over)

## Complex euclidean space

$x\in \mathbb{C}^{N}$ (over)

## Subspaces

Let $V$ be a vector space over $F$ .

A subset $S\subseteq V$ is called a subspace of $V$ if $S$ is a vector space over $F$ in its own right.

$V=\mathbb{R}^{2}$ , $F=\mathbb{R}$ , $S=\text{any line though the origin}$ .

Are there other subspaces?

$S\subseteq V$ is a subspace if and only if for all $a\in F$ and $b\in F$ and for all $s\in S$ and $t\in S$ , $(as+bt)\in S$

## Linear independence

Let ${u}_{1},,{u}_{k}\in V$ .

We say that these vectors are linearly dependent if there exist scalars ${a}_{1},,{a}_{k}\in F$ such that

$\sum_{i=1}^{k} {a}_{i}{u}_{i}=0$
and at least one ${a}_{i}\neq 0$ .

If only holds for the case ${a}_{1}=={a}_{k}=0$ , we say that the vectors are linearly independent .

$1\left(\begin{array}{c}1\\ -1\\ 2\end{array}\right)-2\left(\begin{array}{c}-2\\ 3\\ 0\end{array}\right)+1\left(\begin{array}{c}-5\\ 7\\ -2\end{array}\right)=0$ so these vectors are linearly dependent in $\mathbb{R}^{3}$ .

## Spanning sets

Consider the subset $S=\{{v}_{1}, {v}_{2}, , {v}_{k}\}$ . Define the span of $S$ $\equiv \mathrm{span}(S)\equiv \{\sum_{i=1}^{k} {a}_{i}{v}_{i}\colon {a}_{i}\in F\}$

Fact: $$ is a subspace of $V$ .

$V=\mathbb{R}^{3}$ , $F=\mathbb{R}$ , $S=\{{v}_{1}, {v}_{2}\}$ , ${v}_{1}=\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right)$ , ${v}_{2}=\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right)$ $=\text{xy-plane}$ .

## Aside

If $S$ is infinite, the notions of linear independence and span are easily generalized:

We say $S$ is linearly independent if, for every finite collection ${u}_{1},,{u}_{k}\in S$ , ( $k$ arbitrary) we have $(\sum_{i=1}^{k} {a}_{i}{u}_{i}=0)\implies \forall i\colon {a}_{i}=0$ The span of $S$ is $=\{\sum_{i=1}^{k} {a}_{i}{u}_{i}\colon {a}_{i}\in F\land {u}_{i}\in S\land (k)\}$

In both definitions, we only consider finite sums.

## Bases

A set $B\subseteq V$ is called a basis for $V$ over $F$ if and only if

• $B$ is linearly independent
• $=V$
Bases are of fundamental importance in signal processing. They allow us to decompose a signal into building blocks (basisvectors) that are often more easily understood.

$V$ = (real or complex) Euclidean space, $\mathbb{R}^{N}$ or $\mathbb{C}^{N}$ . $B=\{{e}_{1}, , {e}_{N}\}\equiv \text{standard basis}$ ${e}_{i}=\left(\begin{array}{c}0\\ \\ 1\\ \\ 0\end{array}\right)$ where the 1 is in the $i^{\mathrm{th}}$ position.

$V=\mathbb{C}^{N}$ over. $B=\{{u}_{1}, , {u}_{N}\}$ which is the DFT basis. ${u}_{k}=\left(\begin{array}{c}1\\ e^{-(i\times 2\pi \frac{k}{N})}\\ \\ e^{-(i\times 2\pi \frac{k}{N}(N-1))}\end{array}\right)$ where $i=\sqrt{-1}$ .

## Key fact

If $B$ is a basis for $V$ , then every $v\in V$ can be written uniquely (up to order of terms) in the form $v=\sum_{i=1}^{N} {a}_{i}{v}_{i}$ where ${a}_{i}\in F$ and ${v}_{i}\in B$ .

## Other facts

• If $S$ is a linearly independent set, then $S$ can be extended to a basis.
• If $=V$ , then $S$ contains a basis.

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Abhi
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Salomon
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