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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.
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 $\{a, b, c\}\in F$
,,
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 $v\in V$ :
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:
$x\in \mathbb{R}^{N}$ (over)
$x\in \mathbb{C}^{N}$ (over)
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 $t\in S$ , $(as+bt)\in S$
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
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 $\mathbb{R}^{3}$ .
Consider the subset $S=\{{v}_{1}, {v}_{2}, , {v}_{k}\}$ . Define the span of $S$ $$<S>\equiv \mathrm{span}(S)\equiv \{\sum_{i=1}^{k} {a}_{i}{v}_{i}\colon {a}_{i}\in F\}$$
Fact: $<S>$ is a subspace of $V$ .
$V=\mathbb{R}^{3}$ , $F=\mathbb{R}$ , $S=\{{v}_{1}, {v}_{2}\}$ , ${v}_{1}=\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right)$ , ${v}_{2}=\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right)$ $<S>=\text{xy-plane}$ .
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 $$<S>=\{\sum_{i=1}^{k} {a}_{i}{u}_{i}\colon {a}_{i}\in F\land {u}_{i}\in S\land (k)\}$$∞
A set $B\subseteq V$ is called a basis for $V$ over $F$ if and only if
$V$ = (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 $i=\sqrt{-1}$ .
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 ${v}_{i}\in B$ .
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