MAS1602 - Introductory Algebra

MAS1602 - Introductory Algebra

Vectors and matrices

Complex numbers

Geometry and operations

$z_1z_2$ is analogous to a counterclockwise rotation in the complex plane by $\theta_2$ , and a scalar multiplication by $r_2$

de Moivre's theorem states that \[ z^n = (\cos\theta+i\sin\theta)^n = (\cos(n\theta)+i\sin(n\theta))=e^{in\theta}\]

Complex numbers can be expressed in their modulus-argument or polar form such that\[ x + iy = r(\cos\theta + i\sin\theta)\]
...where $r=|z|=\sqrt{x^2+y^2}$ and $\theta = \arg{z}$

Using the Taylor series of $e^x$, $\sin\theta$ and $\cos\theta$, it can in turn be derived that \[ \cos\theta + i\sin\theta = e^{i\theta} \therefore x+iy\iff re^{i\theta}\]

...where the former is known as Euler's formula

Roots

Any non-zero complex or real number $z$ has $n$ $n$th roots denoted $\zeta_0 \rightarrow \zeta_{n-1}$, where \[\zeta_0=r^{\frac{1}{n}}e^{\frac{i\theta}{n}}\]

The remaining $n-1$ roots can be determined by multiplying $\zeta_0$ by the roots of unity, the complex roots of 1. These are contained in the set \[ \{\zeta_m:\zeta_m^n=1\} \Leftarrow \zeta_m \in \{e^{i\frac{2k\pi}{\theta}}:k=0,1,2,...,n-1\} \]

Vectors

Operations

Dot product

The dot product of two vectors $\underline{u},\underline{v}\in \mathbb{R}^n$ is defined as \[\underline{u}\cdot\underline{v}=u_1v_1+u_2v_2+...+u_nv_n\]

Its properties include

$(\underline{u}\cdot \underline{v})\cdot \underline{w}=\underline{u}\cdot \underline{w} + \underline{v}\cdot \underline{w}$

$\underline{u}\cdot \underline{v}=\underline{v}\cdot \underline{u}$

Cross product

The cross product (or vector product) is exclusively $\in\mathbb{R}^3$ and is defined as $ \underline{u}\times\underline{v}=\begin{pmatrix} u_2v_3-u_3v_2\\ u_3v_1-u_1v_3\\ u_1v_2-u_2v_1 \end{pmatrix} $

$\underline{u}\times\underline{u}=\underline{0}$

$(\underline{v}+\underline{w})\times\underline{u}=(\underline{v}\times\underline{u})+(\underline{w}\times\underline{u})$

$\underline{u}\times(k\underline{v})=k(\underline{u}\times\underline{v})=(k\underline{u})\times\underline{v}$

$\underline{u}\times\underline{v}=-\underline{v}\times\underline{u}$

$\underline{u}\times(\underline{v}+\underline{w})=(\underline{u}\times\underline{v})+(\underline{u}\times\underline{w})$

It can further be shown that $||\underline{u}\times\underline{v}||^2=||\underline{u}||^2||\underline{v}||^2-(\underline{u}\cdot\underline{v})^2$ and by extension \[ ||\underline{u}\times\underline{v}||=||u||\,||v||\sin\theta \]

$\underline{u}\cdot\underline{v}=||\underline{u}||\,||\underline{v}||\cos\theta$

Matrices

Geometry

Planes in $\mathbb{R}^3$ can be determined by $P=\{x\in\mathbb{R}^3:\underline{n}\cdot(\underline{x}-\underline{p_0})=0\}$ where $\underline{p_0}=\begin{pmatrix}x_0\\y_0\\z_0\end{pmatrix}$ is a point on the plane, $\underline{n}=\begin{pmatrix}a\\b\\c\end{pmatrix}$ is a normal vector and $\underline{x}=\begin{pmatrix}x\\y\\z\end{pmatrix}$

Hence the point-normal equation of a plane (or line by negating the $z$ term) is \[ \underline{n}\cdot(\underline{x}-\underline{p_0})=0\iff a(x-x_0)+b(y-y_0)+c(z-z_0)=0 \]

$k(\underline{u}\cdot\underline{v}) = (k\underline{u})\cdot \underline{v} = \underline{u}\cdot (k\underline{v})$

$\underline{u}\times\underline{v}$ is orthogonal to $\underline{u}$ and $\underline{v}$, hence $ \underline{u}\cdot(\underline{u}\times\underline{v})=0 $

Determinant

The cofactor $C_{ij} = (-1)^{i+j}A_{ij}$ (chessboard pattern)

If $B$ is obtained from $A$ by interchanging two rows, then $\det(B)=-\det(A)$

If $B$ is obtained from $A$ by multiplying a row by a constant $k$, then $\det(B)= k\cdot\det(A)$

If $B$ is obtained from $A$ by adding any row to another row, then $\det(B)=\det(A)$

The minor $A_{ij}$ of a matrix $A$ is the determinant of $A$ having deleted row $i$ and column $j$

The determinant of a matrix holds the property that $\det(A)\neq 0\iff \exists\; A^{-1}:A^{-1}A = I_n$

It can be shown that for a fixed $i\in\{1,2,...,n\}$ (namely down any row or column) \[\det(A)=\sum_{j=1}^na_{ij}C_{ij}=\sum_{j=1}^na_{ji}C_{ji}\]

The inverse matrix $A^{-1}$ of a matrix $A$ can be found by performing elementary row operations on an $n\times 2n$ augmented matrix of $A$ and $I_n$ until the left hand side has become $I_n$

The transpose matrix $A^t$ of an $m\times n$ matrix $A$ is an $n\times m$ matrix where $a^t_{ij}=a_{ji}$

$(AB)^t = B^tA^t$

$\underline{v}^t\underline{w} = \underline{v}\cdot \underline{w}$

If $A$ has dimensions $n\times n$, $(A\underline{v})\cdot\underline{w}=\underline{v}\cdot(A^t\underline{w})$

An $n\times n$ matrix is orthogonal if $||Ax||=||x||\iff (A\underline{v})\cdot(A\underline{w})=\underline{v}\cdot\underline{w}$, and holds the property \[AA^t=I_n=A^tA\]

Rotation

For any $\underline{n}=\begin{pmatrix}a\\b\\c\end{pmatrix}$, the infinitesimal rotation matrix $M_{\underline{n}}= \begin{pmatrix} 0 & -c & b \\ c & 0 & - a \\ -b & a & 0 \\ \end{pmatrix} $

$\Rightarrow r_{n,\theta}(\underline{x})=A\underline{x}$ with $A=I_3 + \sin\theta M_n + (1-\cos\theta){M_n}^2$

$r_{\underline{n},\theta}:\mathbb{R}^3\rightarrow\mathbb{R}^3$ denotes the function that rotates any point $\in\mathbb{R}^3$ counterclockwise about line $L$ by $\theta$, where $L$ has equation $\underline{x}=t\underline{n}$

Eigenvectors

A vector $\underline{x}\in\mathbb{R}^n\neq \underline{0}$ is an eigenvector of an $n\times n$ matrix $A$ if $A\underline{x}=\lambda \underline{x}$ with a corresponding eigenvalue $\lambda\in\mathbb{R}$

If $\underline{x}$ is an eigenvector of $A$ then so is $c\underline{x}$ for any $c\in\mathbb{R}\neq 0$

$A\underline{x}=\lambda\underline{x} \Rightarrow \det(A-\lambda I_n)=\underline{0}$ can be used to solve a system for its eigenvalues and hence determine its eigenvectors