\section{Spaces}

\definition{}
A \textit{space} over a field $\mathbb{F}$ consists of the following elements:
\begin{itemize}[itemsep = 2mm]
	\item A set $V$, the elements of which are called \textit{vectors}
	\item An operation called \textit{vector addition}, denoted $+$ \\
		Vector addition operates on two elements of $V$. \\

	\item An operation called \textit{scalar multiplication}, denoted $\times$ \\
		Scalar multiplication multiplies an element of $V$ by an element of $\mathbb{F}$. \\
		Any element of $\mathbb{F}$ is called a \textit{scalar}.
\end{itemize}

\vspace{2mm}

\textbf{Note:}
The same symbols are used for additions and multiplications in both $\mathbb{F}$ and $V$. \\
\textit{These are different operations}, so be aware of the context of each $+$ and $\times$.

\vspace{5mm}

Vector addition and multiplication must have the following properties. \\
In both tables, $x, y, z \in V$ and $a, b\in \mathbb{F}$.

\vspace{2mm}

% [t] and \vspace{0pt} ensure alignment at top
\begin{minipage}[t]{0.48\textwidth}\vspace{0pt}
	\begin{center}
	\begin{tabular}{l | r@{=}l }
		\hline
		\multicolumn{3}{|c|}{Properties of vector addition} \\
		\hline
		Closure			& \multicolumn{2}{c}{$x+y \in V$} \\
		Associativity	& $(x+y)+z~$&$~x+y+z$	\\
		Commutativity	& $x+y~$&$~y+x$ 		\\
		Distributivity	& $x(y+z)~$&$~xy + xz$	\\
		Identity		& $x+0~$&$~x$			\\
		Inverse			& $x + (-x)~$&$~0$
	\end{tabular}
	\end{center}
\end{minipage}%
\hfill%
\begin{minipage}[t]{0.48\textwidth}\vspace{0pt}
	\begin{center}
	\begin{tabular}{l | r@{=}l }
		\hline
		\multicolumn{3}{|c|}{Properties of scalar multiplication} \\
		\hline
		Closure			& \multicolumn{2}{c}{$ax \in V$} \\
		Distributivity	& $a(x+y)~$&$~ax+ay$	\\
						& $(a+b)x~$&$~ax+bx$	\\
		Compatibility$^*$	& $(ab)x~$&$~a(bx)$		\\
		Identity		& $a+0~$&$~a$
	\end{tabular}
\end{center}
\end{minipage}

\vspace{2mm}

$^*$ Remember that $a, b \in \mathbb{F}$ and $x \in V$. Thus, $(ab)$ is multiplication in $\mathbb{F}$ and $(bx)$ is scalar multiplication in $V$. Compatibility is \textit{not} associativity. \\

Likewise, the addition you see in the distributive property of multiplication is field addition, not vector addition.

\vspace{6mm}

Usually, the word \textit{vector} refers to an element of $\mathbb{R}^n$. As you might expect $\mathbb{R}^n$ is a vector space over the field $\mathbb{R}$ under our usual vector operations.

Here's a quick review of these operations:
\begin{itemize}
	\item Scalar multiplication is done elementwise: $3 \times [a, b, c] = [3a, 3b, 3c]$.
	\item Vector addition is similar: $[a, b, c] + [1, 2, 3] = [a+1,~b+2,~c+3]$.
	\item Vector addition is not valid for vectors of different sizes.
\end{itemize}

\problem{}
Verify that $\mathbb{R}^n$ is a vector space over $\mathbb{R}$ under these operations.

\vfill

\pagebreak

We can also define an \textit{inner product} or \textit{vector product} that takes two elements of $V$ and produces another. \\

When we work in $\mathbb{R}^n$, we usually use the dot product as our vector product. It is defined as follows: \\

\definition{Dot Product}
Given two vectors $a, b \in \mathbb{R}^n$, the  \textit{dot product} of $a$ and $b$ (written $a \cdot b$ or $\langle a, b \rangle$) is $\sum_1^n a_ib_i$.

\vspace{2mm}

For example, if $a = [1, 2, 3]$ and $b = [4, 5, 6]$,
$$
	\langle a, b \rangle = (1 \times 4) + (2 \times 5) + (3 \times 6) = 32
$$
As you may expect, the dot product $\langle a, b \rangle$ is valid iff $a$ and $b$ are the same size.



\problem{}
Show that the dot product is commutative.

\vfill

\problem{}
Show that the dot product is positive-definite. \\
This means that $\langle a, a \rangle > 0$ unless $a = 0$.

\vfill
\pagebreak