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.