definitions

algebra terms

a set is a collection of elements:

$$S=\{a,b,c,\dots \}$$

a binary operation on $S$ is a map

$$\ast :S\times S\to S.$$

a group is a pair $(G,\ast )$, where $G$ is a set and $\ast$ is a binary operation satisfying, for all $a,b,c\in G$:

$$a\ast b\in G,(a\ast b)\ast c=a\ast (b\ast c),$$

and there is an identity $e\in G$ and inverse $a^{-1}\in G$ such that

$$e\ast a=a\ast e=a,a\ast a^{-1}=e.$$

a group is abelian if it is commutative:

$$a\ast b=b\ast a.$$

a field is a set $F$ with operations $+$ and $\cdot$, written $(F,+,\cdot )$, where $(F,+)$ and $(F\setminus \{0\},\cdot )$ are abelian groups and multiplication distributes over addition:

$$a(b+c)=ab+ac.$$

a vector space $V$ over a field $F$ has vector addition and scalar multiplication:

$$+:V\times V\to V,\cdot :F\times V\to V.$$

if $u,v\in V$ and $\lambda \in F$, then

$$u+v\in V,\lambda v\in V.$$

a vector is an element of a vector space: $v\in V$.

$$\text{group}⟶\text{field}⟶\text{vector space}⟶\text{vector}$$

example: $\mathbb{R}$ is a field, ${\mathbb{R}}^n$ is a vector space over $\mathbb{R}$, and $(v_1,\dots ,v_n)\in {\mathbb{R}}^n$ is a vector.

systems of linear equations

a system of $m$ linear equations in $n$ unknowns can be written as

$$\left(\begin{array}{c}{a}_{11}\\ ⋮\\ a_{m1}\end{array}\right)x_1+\left(\begin{array}{c}{a}_{12}\\ ⋮\\ a_{m2}\end{array}\right)x_2+\cdots +\left(\begin{array}{c}{a}_{1n}\\ ⋮\\ a_{mn}\end{array}\right)x_n=\left(\begin{array}{c}{b}_1\\ ⋮\\ b_m\end{array}\right).$$

equivalently,

$$Ax=b,$$

where

$$A=\left(\begin{array}{cccc}{a}_{11}& a_{12}& \cdots & a_{1n}\\ a_{21}& a_{22}& \cdots & a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{m1}& a_{m2}& \cdots & a_{mn}\end{array}\right)\in {\mathbb{R}}^{m\times n},$$ $$x=\left(\begin{array}{c}{x}_1\\ x_2\\ ⋮\\ x_n\end{array}\right)\in {\mathbb{R}}^{n\times 1},b=\left(\begin{array}{c}{b}_1\\ b_2\\ ⋮\\ b_m\end{array}\right)\in {\mathbb{R}}^{m\times 1}.$$

the augmented matrix is

$$[A\mid b]=\left[\begin{array}{ccccc}{a}_{11}& a_{12}& \cdots & a_{1n}& b_1\\ a_{21}& a_{22}& \cdots & a_{2n}& b_2\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ a_{m1}& a_{m2}& \cdots & a_{mn}& b_m\end{array}\right].$$

the entry $a_{ij}$ is in row $i$, column $j$. the dimensions match because

$$(m\times n)(n\times 1)=m\times 1.$$

matrix facts

matrix multiplication is distributive and associative, but generally not commutative:

$$AB\ne BA.$$

the inverse of $A$, when it exists, is $A^{-1}$, with

$$A{A}^{-1}=I=A^{-1}A.$$

the transpose of $A$ is $A^T$. a matrix is symmetric if $A=A^T$.

useful identities:

$$(A^{-1}{)}^T=(A^T{)}^{-1},(AB{)}^T=B^T{A}^T,(AB{)}^{-1}=B^{-1}{A}^{-1},$$ $$(A+B{)}^T=A^T+B^T,(A+B{)}^{-1}\ne A^{-1}+B^{-1}\text{ in general.}$$