There are four main components to a vector space: 1. A set $V$ 2. A field $\mathbb{F}$ 3. An operation “$+$” that allows us to add together elements of $V$ 4. An operation “$\cdot$” that allows us to multiply, or scale, an element of $V$

Notation ($V\times W$)

Given two sets $V$ and $W$, we can make a new set $V\times W$, which we would call the product of $V$ and $W$, that is: $$\begin{align*} V\times W:=\{(v,w)|v\in V,w\in W\} \end{align*}$$ or that $V\times W$ produces a new set of pairs of elements where the first element is all possible values of $V$ and the same for the second and $W$.

Definition (Vector Space)

An $\mathbb{F}$-vector space is a set $V$, together with an addition operation $$\begin{align*} V \times V &\longrightarrow V\\ (v,w) &\longmapsto v+w\\ \end{align*}$$ and a scalar multiplication operation $$\begin{align*} \mathbb{F} \times V &\longrightarrow V\\ (c,v)&\longmapsto c\cdot v \end{align*}$$ (i.e. defined as $(V,+,\cdot)$) that satisfy the following six rules: 1. Commutativity: If $v,w \in V$ then $$\begin{align*}v+w=w+v\end{align*}$$ 2. Associativity: If $u,v,w\in V$ and $a,b\in\mathbb{F}$ then $$\begin{align*} (u+v)+w&=u+(v+w) \newline (ab)\cdot v&=a\cdot (b\cdot v) \end{align*}$$ 3. Additive identity: There is an element $0\in V$ such that for $v\in V$, $$0+v=v$$ 4. Additive inverse: For every $v\in V$ there is a $w\in W$ such that $$v+w=0$$ 5. Multiplicative identity: For all $v\in V$, $$\begin{align*} 1\cdot v=v \end{align*}$$ 6. Distributivity: Given $a,b\in \mathbb{F}$ and $v,w\in V$, we have: $$\begin{align*} a\cdot(v+w)&=a\cdot v+a\cdot w \newline (a+b)\cdot v&=a\cdot v+b\cdot v \end{align*}$$

Notation (Set of Functions)

Let $S$ be any set. We use the notation $\mathbb{F}^S$ to denote the set of function from $S$ to $\mathbb{F}$.

Example: Sequences of real numbers: $(a_k)_{k=0}^\infty$, this describes a function $f:\mathbb{N}\mapsto\mathbb{R}$ where $f(k)=a_k$. So $(a_k)_{k=0}^\infty$ can be considered an element of $\mathbb{R}^\mathbb{N}$.

Notation (Polynomials)

A polynomial of one variable, with coefficients in $\mathbb{F}$, is an expression of the form $$\begin{align*} p(t)=a_nt^n+a_{n-1}t^{n-1}+...+a_1t+a_o \end{align*}$$ where $a_i\in\mathbb{F}$, $\forall i=0,1,...n$. The degree of $p$, denoted $deg(p$), is given by $$\begin{align*} deg(p)=max\{k|a_k\not=0\} \end{align*}$$ We use $\mathbb{F}[t]$ to denote the set of all polynomial with coefficients in $\mathbb{F}$, and we use $\mathbb{F}[t]_{\leq n}$ to denoted the set of all polynomials with degree $\leq n$.