\[\newcommand{\u}[1]{\boldsymbol{\mathsf{#1}}} \renewcommand{\b}[1]{\boldsymbol{#1}} \newcommand{\t}[1]{\textsf{#1}} \newcommand{\m}[1]{\mathbb{#1}} \def\RR{\bf R} \def\bold#1{\bf #1} \def\mbf#1{\mathbf #1} \def\uv#1{\hat{\usf {#1}}} \def\dl#1{\underline{\underline{#1}}} \newcommand{\usf}[1]{\boldsymbol{\mathsf{#1}}} \def\bs#1{\usf #1}\]

A real vector space

\[\newcommand{\b}[1]{\boldsymbol{#1}} \newcommand{\pr}[1]{\left(#1\right)}\]

A real vector space is a set \(\mathbb{V}\) on which the two operations of $\cdot + \cdot : \mathbb{V}\times \mathbb{V} \to \mathbb{V}$ and $\cdot * \cdot: \mathbb{V}\times \mathbb{R}\to \mathbb{R}$ are defined. There exists a zero element in $\mathbb{V}$, which we term $\b{0}_{\mathbb{V}}$, or simply as $\b{0}$ when there is no confusion. For every $\b{u}\in \mathbb{V}$,there exists $\b{u}’$. These operation and elements need to satisfy the following properties

For $\b{u}$, $\b{v}$, $\b{z}\in \mathbb{V}$ we have that

1. $\b{u}+\b{v}$=$\b{v}+\b{u}$ (symmetry/commutative)
2. $\b{u}+(\b{v}+\b{z})$=$( \b{u}+\b{v})+\b{z}$ (associative)
3. $\b{u}+\u{0}$=$\b{u}$
4. $\b{u}+\b{u}’=\u{0}$.

For every $a,b\in \mathbb{R}$ and $\b{u}\in \mathbb{V}$ we have that

1. $a*\pr{b \b{u}}=\pr{ab} * \b{u}$
2. $(a+b) * \b{u}=a * \b{u}+b * \b{u}$
3. $a * \pr{\b{u}+\b{v}}=a * \b{u}+a * \b{v}$
4. $1 * \b{u}=\b{u}$

We can show identities such as $0 * \b{u}=\b{0}$, and $-1 * \b{u}=\b{u}’$ from the above axiom.

\[\begin{align} \pr{1+0} * \b{u}&=1 * \b{u}+(0) * \b{u}\\ \pr{1} * \b{u}&=1 * \b{u}+(0) * \b{u}\\ \b{u}&=\b{u}+(0) * \b{u}\\ \b{0}&=(0)*\b{u}\\ \end{align}\] \[\begin{align} \pr{1+(-1)}\b{u}&=1*\b{u}+(-1)*\b{u}\\ \pr{0}\b{u}&=1*\b{u}+(-1)*\b{u}\\ \b{0}&=1*\b{u}+(-1)*\b{u}\\ \b{0}&=\b{u}+(-1)*\b{u}\\ (-1) * \b{u}&=\b{u}' \end{align}\]

When there is no confusion we will skip explicity writing $*$, so expressions such as $a * \pr{b \b{u}}=\pr{ab} * \b{u}$ will appears as $a\pr{b \b{u}}=\pr{ab} \b{u}$