1.5 Review of linear algebra  (Page 2/2)

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Dimension

Let $V$ be a vector space with basis $B$ . The dimension of $V$ , denoted $\mathrm{dim}(V)$ , is the cardinality of $B$ .

Every vector space has a basis.

Every basis for a vector space has the same cardinality.

$\implies \mathrm{dim}(V)$ is well-defined .

If $\mathrm{dim}(V)$ , we say $V$ is finite dimensional .

Examples

vector space field of scalars dimension
$\mathbb{R}^{N}$ $\mathbb{R}$
$\mathbb{C}^{N}$ $\mathbb{C}$
$\mathbb{C}^{N}$ $\mathbb{R}$

Every subspace is a vector space, and therefore has its own dimension.

Suppose $(S=\{{u}_{1}, , {u}_{k}\})\subseteq V$ is a linearly independent set. Then $\mathrm{dim}()=$

Facts

• If $S$ is a subspace of $V$ , then $\mathrm{dim}(S)\le \mathrm{dim}(V)$ .
• If $\mathrm{dim}(S)=\mathrm{dim}(V)$ , then $S=V$ .

Direct sums

Let $V$ be a vector space, and let $S\subseteq V$ and $T\subseteq V$ be subspaces.

We say $V$ is the direct sum of $S$ and $T$ , written $V=(S, T)$ , if and only if for every $v\in V$ , there exist unique $s\in S$ and $t\in T$ such that $v=s+t$ .

If $V=(S, T)$ , then $T$ is called a complement of $S$ .

$V={C}^{}=\left\{f:\mathbb{R}\mathbb{R}|f\text{is continuous}\right\}$ $S=\text{even funcitons in}{C}^{}$ $T=\text{odd funcitons in}{C}^{}$ $f(t)=\frac{1}{2}(f(t)+f(-t))+\frac{1}{2}(f(t)-f(-t))$ If $f=g+h={g}^{}+{h}^{}$ , $g\in S$ and ${g}^{}\in S$ , $h\in T$ and ${h}^{}\in T$ , then $g-{g}^{}={h}^{}-h$ is odd and even, which implies $g={g}^{}$ and $h={h}^{}$ .

Facts

• Every subspace has a complement
• $V=(S, T)$ if and only if
• $S\cap T=\{0\}$
• $=V$
• If $V=(S, T)$ , and $\mathrm{dim}(V)$ , then $\mathrm{dim}(V)=\mathrm{dim}(S)+\mathrm{dim}(T)$

Invoke a basis.

Norms

Let $V$ be a vector space over $F$ . A norm is a mapping $(V, F)$ , denoted by $()$ , such that forall $u\in V$ , $v\in V$ , and $\in F$

• $(u)> 0$ if $u\neq 0$
• $(u)=\left|\right|(u)$
• $(u+v)\le (u)+(v)$

Examples

Euclidean norms:

$x\in \mathbb{R}^{N}$ : $(x)=\sum_{i=1}^{N} {x}_{i}^{2}^{\left(\frac{1}{2}\right)}$ $x\in \mathbb{C}^{N}$ : $(x)=\sum_{i=1}^{N} \left|{x}_{i}\right|^{2}^{\left(\frac{1}{2}\right)}$

Induced metric

Every norm induces a metric on $V$ $d(u, v)\equiv (u-v)$ which leads to a notion of "distance" between vectors.

Inner products

Let $V$ be a vector space over $F$ , $F=\mathbb{R}$ or $\mathbb{C}$ . An inner product is a mapping $V\times VF$ , denoted $\dot$ , such that

• $v\dot v\ge 0$ , and $(v\dot v=0, v=0)$
• $u\dot v=\overline{v\dot u}$
• $au+bv\dot w=a(u\dot w)+b(v\dot w)$

Examples

$\mathbb{R}^{N}$ over: $x\dot y=x^Ty=\sum_{i=1}^{N} {x}_{i}{y}_{i}$

$\mathbb{C}^{N}$ over: $x\dot y=(x)y=\sum_{i=1}^{N} \overline{{x}_{i}}{y}_{i}$

If $(x=\left(\begin{array}{c}{x}_{1}\\ \\ {x}_{N}\end{array}\right))\in \mathbb{C}$ , then $(x)\equiv \left(\begin{array}{c}\overline{{x}_{1}}\\ \\ \overline{{x}_{N}}\end{array}\right)^T$ is called the "Hermitian," or "conjugatetranspose" of $x$ .

Triangle inequality

If we define $(u)=u\dot u$ , then $(u+v)\le (u)+(v)$ Hence, every inner product induces a norm.

Cauchy-schwarz inequality

For all $u\in V$ , $v\in V$ , $\left|u\dot v\right|\le (u)(v)$ In inner product spaces, we have a notion of the angle between two vectors: $((u, v)=\arccos \left(\frac{u\dot v}{(u)(v)}\right))\in \left[0 , 2\pi \right)$

Orthogonality

$u$ and $v$ are orthogonal if $u\dot v=0$ Notation: $(u, v)$ .

If in addition $(u)=(v)=1$ , we say $u$ and $v$ are orthonormal .

In an orthogonal (orthonormal) set , each pair of vectors is orthogonal (orthonormal).

Orthonormal bases

An Orthonormal basis is a basis $\{{v}_{i}\}$ such that ${v}_{i}\dot {v}_{i}={}_{ij}=\begin{cases}1 & \text{if i=j}\\ 0 & \text{if i\neq j}\end{cases}$

The standard basis for $\mathbb{R}^{N}$ or $\mathbb{C}^{N}$

The normalized DFT basis ${u}_{k}=\frac{1}{\sqrt{N}}\left(\begin{array}{c}1\\ e^{-(i\times 2\pi \frac{k}{N})}\\ \\ e^{-(i\times 2\pi \frac{k}{N}(N-1))}\end{array}\right)$

Expansion coefficients

If the representation of $v$ with respect to $\{{v}_{i}\}$ is $v=\sum {a}_{i}{v}_{i}$ then ${a}_{i}={v}_{i}\dot v$

Gram-schmidt

Every inner product space has an orthonormal basis. Any (countable) basis can be made orthogonal by theGram-Schmidt orthogonalization process.

Orthogonal compliments

Let $S\subseteq V$ be a subspace. The orthogonal compliment $S$ is ${S}^{}=\{u\colon u\in V\land (u\dot v=0)\land \forall v\colon v\in S\}$ ${S}^{}$ is easily seen to be a subspace.

If $\mathrm{dim}(v)$ , then $V=(S, {S}^{})$ .

If $\mathrm{dim}(v)$ , then in order to have $V=(S, {S}^{})$ we require $V$ to be a Hilbert Space .

Linear transformations

Loosely speaking, a linear transformation is a mapping from one vector space to another that preserves vector space operations.

More precisely, let $V$ , $W$ be vector spaces over the same field $F$ . A linear transformation is a mapping $T:VW$ such that $T(au+bv)=aT(u)+bT(v)$ for all $a\in F$ , $b\in F$ and $u\in V$ , $v\in V$ .

In this class we will be concerned with linear transformations between (real or complex) Euclidean spaces , or subspaces thereof.

Image

$()$ T w w W T v w for some v

Nullspace

Also known as the kernel: $\mathrm{ker}(T)=\{v\colon v\in V\land (T(v)=0)\}$

Both the image and the nullspace are easily seen to be subspaces.

Rank

$\mathrm{rank}(T)=\mathrm{dim}(())$ T

Nullity

$\mathrm{null}(T)=\mathrm{dim}(\mathrm{ker}(T))$

Rank plus nullity theorem

$\mathrm{rank}(T)+\mathrm{null}(T)=\mathrm{dim}(V)$

Matrices

Every linear transformation $T$ has a matrix representation . If $T:{𝔼}^{N}{𝔼}^{M}$ , $𝔼=\mathbb{R}$ or $\mathbb{C}$ , then $T$ is represented by an $M\times N$ matrix $A=\begin{pmatrix}{a}_{11} & & {a}_{1N}\\ & & \\ {a}_{M1} & & {a}_{MN}\\ \end{pmatrix}$ where $\left(\begin{array}{c}{a}_{1i}\\ \\ {a}_{Mi}\end{array}\right)=T({e}_{i})$ and ${e}_{i}=\left(\begin{array}{c}0\\ \\ 1\\ \\ 0\end{array}\right)$ is the $i^{\mathrm{th}}$ standard basis vector.

A linear transformation can be represented with respect to any bases of $𝔼^{N}$ and $𝔼^{M}$ , leading to a different $A$ . We will always represent a linear transformation using the standard bases.

Column span

$\mathrm{colspan}(A)==()$ A

Duality

If $A:{\mathbb{R}}^{N}{\mathbb{R}}^{M}$ , then $\mathrm{ker}(A)^{}=()$ A

If $A:{\mathbb{C}}^{N}{\mathbb{C}}^{M}$ , then $\mathrm{ker}(A)^{}=()$ A

Inverses

The linear transformation/matrix $A$ is invertible if and only if there exists a matrix $B$ such that $AB=BA=I$ (identity).

Only square matrices can be invertible.

Let $A:{𝔽}^{N}{𝔽}^{N}$ be linear, $𝔽=\mathbb{R}$ or $\mathbb{C}$ . The following are equivalent:

• $A$ is invertible (nonsingular)
• $\mathrm{rank}(A)=N$
• $\mathrm{null}(A)=0$
• $\det A\neq 0$
• The columns of $A$ form a basis.

If $A^{(-1)}=A^T$ (or $(A)$ in the complex case), we say $A$ is orthogonal (or unitary ).

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Ali
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da
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Bhagvanji
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Giriraj
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Damian
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Professor
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Professor
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Kyle
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biomolecules are e building blocks of every organics and inorganic materials.
Joe
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