Vector space and subspace 

 A set $\mathcal{W}$  is called a vector space if its elements are vectors and 

• the sum of two elements of $\mathcal{W}$ is again an element of $\mathcal{W}$, and
• the product of an element of $\mathcal{W}$ with a scalar is an element of $\mathcal{W}$.  

A subset of a vector space which itself is a vector space is called a subspace. 

Span 

Let $a_{i} \in \mathcal{W}$  $i=1, \ldots, n$, where $\mathcal{W}$ is a vector space. The set of all linear combinations of $a_{1}, \ldots, a_{n}$, denoted as $\{ a_{1}, \ldots, a_{n} \}$, is a subspace of $\mathcal{W}$: $\{ a_{1}, \ldots, a_{n} \} \subset \mathcal{W}$. If every vector of a vector space $\mathcal{V}$ can be written as a linear combination of $a_{1}, \ldots, a_{n}$, then $a_{1}, \ldots, a_{n}$ is said to span $\mathcal{V}$: $\mathcal{V} = \{ a_{1}, \ldots, a_{n} \}$.

Basis and dimension of a vector space

A basis of a vector space $\mathcal{W}$ is a set of linear independent vectors which span $\mathcal{W}$.

Every vector space contains a basis and every vector can be written as a unique linear combination of the vectors of a basis.

The dimension of a vector space $\mathcal{W}$, denoted as $\dim W$, is the number of vectors of a basis of $\mathcal{W}$. In an $n$-dimensional vector space $\mathcal{W}$, every linear independent set of $n$ vectors is a basis of $\mathcal{W}$. 

As an example, the three-dimensional space $\mathbb{R}^3$ is a vector space, and one possible basis for $\mathbb{R}^3$  is a set of the following unit vectors: $$\begin{bmatrix}1\\0\\0 \end{bmatrix}, ~~~ \begin{bmatrix}0\\1\\0 \end{bmatrix}, ~~~ \begin{bmatrix}0\\0\\1 \end{bmatrix}.$$

That is all the vectors in  $\mathbb{R}^3$ can be written as linear combination of these three unit vectors. For example, the arbitrary three dimentional  vector $\begin{bmatrix} 4\\3\\5 \end{bmatrix}$ can be written as the linear combination:

$$\begin{bmatrix} 4\\3\\5 \end{bmatrix}=4\begin{bmatrix}1\\0\\0 \end{bmatrix}+3\begin{bmatrix}0\\1\\0 \end{bmatrix}+5\begin{bmatrix}0\\0\\1 \end{bmatrix}.$$ We can say the three vectors  $\begin{bmatrix}1\\0\\0 \end{bmatrix}, \begin{bmatrix}0\\1\\0 \end{bmatrix}$, and $\begin{bmatrix}0\\0\\1 \end{bmatrix}$span the vector space $\mathbb{R}^3$ . 

Column Space (or Range Space) of a Matrix

The column space (or range space) of a matrix $A$ of size  $m\times n$, is the subspace of $\mathbb{R}^{m}$ which is spanned by the column vectors of $A$. The range space of $A$ is denoted as $\mathcal{R}(A)$.  The dimension of $\mathcal{R}(A)$, equals the maximum number of linear independent column vectors of $A$.

For example, let $A=\begin{bmatrix} 1&1\\1&2\\1&3 \end{bmatrix}$. Then the $\mathcal{R}(A)$ is a subspace in $\mathbb{R}^{3}$, and its elements can be written as linear combination of the two column vectors of $A$:

$$\begin{bmatrix} 1\\1\\1 \end{bmatrix}, ~ \text{and} ~ \begin{bmatrix} 1\\2\\3 \end{bmatrix}.$$ In this example, the dimension of $\mathcal{R}(A)$ is 2 becouse $A$ has two independent columns.