A* vector space* $V$ is a set of objects which can be added and multiplied by numbers, in such a way that the sum of two elements of $V$ is again an element of $V$, the product of an element of $V$ by a number is an element of $V$, and the following properties are satisfied:

VS 1. Given $u,v,w\in V$, we have

$$(u+v)+w=u+(v+w).$$

VS 2. There is an element $O\in V$ such that

$$O+u=u+O=u$$

for all $u\in V$.

VS 3. Given $u\in V$, the element $(-1)u$ is such that

$$u+(-1)u=(-1)u+u=O.$$

$(-1)u$ is simply written as $-u$.

VS 4. For all $u,v\in V$, we have

$$u+v=v+u.$$

VS 5. If $c$ is a number, then $c(u+v)=cu+cv$.

VS 6. If $a,b$ are two numbers, then $(a+b)v=av+bv$.

VS 7. If $a,b$ are two numbers, then $(ab)v=a(bv)$.

VS 8. For any $u\in V$, we have $1u=u$.

The axioms VS 1-VS 4 say that $(V,+)$ is an abelian group. The elements of vector space $V$ are called *vectors*. One can easily verify that vectors in $\mathbb{R}^n$ satisfy the axioms VS 1-VS 8 and hence $\mathbb{R}^n$ is a vector space.

*Example*. Let $M(m,n)$ denote the set of all $m\times n$ matrices. Then $M(m,n)$ is a vector space. Using the identification

$$\begin{pmatrix}a_{11} & a_{12} & \cdots & a_{1n}\\a_{21} & a_{22} & \cdots & a_{2n}\\\vdots &\vdots& \ddots&\vdots\\a_{m1} & a_{m2} & \cdots & a_{mn}\end{pmatrix}\longleftrightarrow(a_{11},\cdots,a_{1n};a_{21},\cdots,a_{2n};\cdots;a_{m1},\cdots,a_{mn}),$$

we see that $M(m,n)$ may be identified with $\mathbb{R}^{mn}$ as a vector space.

*Example*. Let $\mathcal{F}$ be the set of all functions from $\mathbb{R}$ to $\mathbb{R}$. For any $f,g\in\mathcal{F}$, define $f+g$ by

$$(f+g)(x)=f(x)+g(x)$$

for all $x\in\mathbb{R}$. For any $f\in\mathbb{R}$ and for any number $c$, define $cf$ by

$$(cf)(x)=cf(x)$$

for all $x\in\mathbb{R}$. Then $\mathcal{F}$ is a vector space called a *function space*.

*Example*. Let $V=\{ae^t+be^{2t}: a,b\in\mathbb{R}\}$. Then $V$ is a vector space. Note that $V$ is the set of all solutions of the second order linear differential equation $\frac{d^2x}{dt^2}-3\frac{dx}{dt}+2x=0$.

**Subspaces**

A subset $U$ of a vector space $V$ is said to be a *subspace* if $U$ itself is also a vector space. For $U$ to be a vector space, it suffices to satisfy that

(i) For any $v,w\in U$, $v+w\in U$.

(ii) If $v\in U$ and $c$ is a number, $cv\in U$.

(iii) The identity element $O$ of $V$ is also an element of $U$.

*Proposition*. A nonempty subset $U$ of a vector space $V$ is a subspace if and only if $av+bw\in U$ for any $v,w\in U$ and numbers $a,b$.

*Proof*. Exercise

*Example*. Let $U$ be the set of vectors in $\mathbb{R}^n$ whose last coordinate is $0$. Then $U$ is a subspace of $\mathbb{R}^n$. $U$ may be identified with $\mathbb{R}^{n-1}$.

*Example*. Let $A$ be a vector in $\mathbb{R}^n$. Let $U$ be the set of all vectors $B$ in $\mathbb{R}^n$ such that $B\cdot A=0$ i.e. $B$ is perpendicular to $A$. Then $U$ is a subspace of $V$.

Example. Let $U$ and $W$ be subspaces of a vector space $V$. Then $U\cap W$ is also a subspace of $V$.

*Example*. Let $U$ and $W$ be subspaces of a vector space $V$. Define the sum of $U$ and $W$

$$U+W=\{u+w: u\in U\ \rm{and}\ w\in W\}.$$

Then $U+W$ is a subspace.