# Review of linear algebra

 Page 2 / 2

## 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 ).

can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
im not good at math so would this help me
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
how do you translate this in Algebraic Expressions
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!