Vectors

• magnitudes in directions
• ordered tuples
• point in space

$${\mathbb{R}}^n=\left\{\left(\begin{array}{c}{x}_1\\ x_2\\ \dots \\ x_n\end{array}\right)\right\}$$

Operations §

Addition §

[!math|{“type”:“definition”,“number”:“auto”,“setAsNoteMathLink”:false,“title”:“Addition”,“label”:“addition”,“_index”:0}] Definition 1 (Addition).

For

\vec{v}=\begin{pmatrix} x_{1} \\ y_{1} \end{pmatrix}, \vec{w}=\begin{pmatrix} x_{2} \\ y_{2} \end{pmatrix} ,$$

\vec{v}+\vec{w}=\begin{pmatrix} x_{1}+x_{2} \ y_{1}+y_{2} \end{pmatrix} .$$

Intuition: joining two arrows from tail to head

Scalar Multiplication §

[!math|{“type”:“definition”,“number”:“auto”,“setAsNoteMathLink”:false,“title”:“Scalar Multiplication”,“label”:“scalar-multiplication”,“_index”:1}] Definition 2 (Scalar Multiplication).

\alpha\cdot \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} \alpha x \\ \alpha y \end{pmatrix} .$$

Intuition: scaling arrow in direction by $\alpha$.

Binary Operation §

[!math|{“type”:“definition”,“number”:“auto”,“setAsNoteMathLink”:false,“title”:“BInary Operation”,“label”:“binary-operation”,“_index”:2}] Definition 3 (BInary Operation). For set $S$, binary operation $\ast$: $S\times S\to S$ Examples: $\times ,+$ on $\mathbb{R}$, $\circ$ on $\{f:\mathbb{R}\to \mathbb{R}\}$

[!math|{“type”:“definition”,“number”:“auto”,“setAsNoteMathLink”:false,“title”:“Commutative”,“label”:“commutative”,“_index”:3}] Definition 4 (Commutative). $\ast$ is commutative if $a\ast b=b\ast a$ for $a,b\in S$. Ex: $\times ,+$ are commutative, $\circ$ is not.

[!math|{“type”:“definition”,“number”:“auto”,“setAsNoteMathLink”:false,“title”:“Associative”,“label”:“associative”,“_index”:4}] Definition 5 (Associative). $(a\ast b)\ast c=a\ast (b\ast c)$ for all $a,b,c\in S$ Example: $+,\circ ,\times$ are all associative.

Identity Element §

[!math|{“type”:“definition”,“number”:“auto”,“setAsNoteMathLink”:false,“title”:“Identity Element”,“label”:“identity-element”,“_index”:5}] Definition 6 (Identity Element). $e\in S$ is identity element for $\ast$ if

$$e\ast s=s\ast e=s$$

for all $s\in S$. Examples: $f(x)=x$ is identity element of $S=\{f:\mathbb{R}\to \mathbb{R}\}$

[!math|{“type”:“proposition”,“number”:“auto”,“setAsNoteMathLink”:false,“title”:“Unique Identity”,“label”:“unique-identity”,“_index”:6}] Proposition 7 (Unique Identity). Identity elements are unique.

\begin{proof}Suppose $e,f$ are both identities. Then,

$$e=e\ast f=f$$

\end{proof}

Inverses §

[!math|{“type”:“definition”,“number”:“auto”,“setAsNoteMathLink”:false,“title”:“Inverse”,“label”:“inverse”,“_index”:7}] Definition 8 (Inverse). For $\ast$ with identity $e$, then $b\in S$ is inverse to $a\in S$ if $a\ast b=b\ast a=e$.

Inverses for Composition §

$$g=f^{-1}⟺f(g(x))=x\wedge g(f(x))=x.$$

• to solve $g(f(x))=x$, need $f$ is injective
  • then set $g(f(x))=x$ for all $x$, $g(y)=0$ if $y$
• to solve $f(g(x))=x$, need $f$ is surjective