Span for vector space I want to know why the set S={\begin{bmatrix}1 & 0 \\0 & 1 \end{bmatrix},\begin{bmatrix}0 & 1 \\1 & 0 \end{bmatrix}} does not span   M  2 × 2   ?
 A: To span $\mathbb{M}_{2\times2}$ (which I understand as the group of $2\times2$ matrices), every matrix must be a linear combination of the matrices in the spanning set.
In your case, for example:
$$\begin{pmatrix} 1&2\\3&4\end{pmatrix}$$
cannot be achieved.
A: If I assume $\mathbb{M}_{2\times2}$ is a vector space of $2\times 2$ matrix then any element of $\mathbb{M}_{2\times2}$ will be of the form 
$\begin{bmatrix}a & b \\c & d \end{bmatrix}$. To be A spanning set of vector space means every element of the vector space can be expressed as a linear combination of elements of the given set. In this case spanning set of any $2\times 2 $ matrix can be given by the set
$S=\{\begin{bmatrix}1 & 0 \\0 & 0 \end{bmatrix}, \begin{bmatrix}0 & 1 \\0 & 0 \end{bmatrix}, \begin{bmatrix}0 & 0\\1 & 0 \end{bmatrix}, \begin{bmatrix}0 & 0 \\0 & 1\end{bmatrix}\}$. 
As any matrix $\begin{bmatrix}a & b \\c & d \end{bmatrix}$ $\in $$\mathbb{M}_{2\times2}$ can be written as a  linear combination of matrices of $S$ as follows
$\begin{bmatrix}a & b \\c & d \end{bmatrix}=a\begin{bmatrix}1 & 0  \\0 & 0 \end{bmatrix}+ b\begin{bmatrix}0 & 1 \\0 & 0 \end{bmatrix} +c\begin{bmatrix}0 & 0\\1 & 0 \end{bmatrix}+ d\begin{bmatrix}0 & 0 \\0 & 1\end{bmatrix}$.
Jon Mark answer tells you that you can't write matrix $\begin{pmatrix} 1&2\\3&4\end{pmatrix}$ as a linear combination of matrices $\begin{bmatrix}1 & 0 \\0 & 1 \end{bmatrix}$ and $\begin{bmatrix}0 & 1 \\1 & 0 \end{bmatrix}$.
