How to find a basis for $2\times 2$ matrix Consider $W$ the subset of the Vector space $V$ where $V$ is all 2x2 matrices:
$$
W = \left\{ \begin{pmatrix} a & a \\ a & b \end{pmatrix} \mid a,b \in \mathbb{R} \right\}
$$
How would I find a basis for $W$?
 A: Hint: Every $2\times 2$ matrix may be written as
$$
\begin{bmatrix}
a & b\\ c& d
\end{bmatrix}
=
a\begin{bmatrix}
1 & 0\\ 0 & 0
\end{bmatrix}
+
b\begin{bmatrix}
0 & 1\\ 0 & 0
\end{bmatrix}
+
c\begin{bmatrix}
0 & 0\\ 1 & 0
\end{bmatrix}
+d\begin{bmatrix}
0 & 0\\ 0 & 1
\end{bmatrix}
$$
This shows that
$$
\left\{
\begin{bmatrix}
1 & 0\\ 0 & 0
\end{bmatrix}
,\begin{bmatrix}
0 & 1\\ 0 & 0
\end{bmatrix}
,\begin{bmatrix}
0 & 0\\ 1 & 0
\end{bmatrix}
,\begin{bmatrix}
0 & 0\\ 0 & 1
\end{bmatrix}
\right\}
$$
is a basis for the vector space $M_2$ of all $2\times 2$ matrices. 
Can you construct a similar argument for your subspace $W$?
A: You've already given us a parametrization in two variables, $a$ and $b$. What you've essentially told us is that you are looking for matrices of the form
$$\begin{pmatrix} a&a \\ a&b\end{pmatrix} = a\begin{pmatrix} 1&1 \\ 1&0\end{pmatrix} + b\begin{pmatrix} 0&0\\0&1\end{pmatrix},$$
so that $\begin{pmatrix} 1&1\\1&0\end{pmatrix}$ and $\begin{pmatrix} 0&0\\0&1\end{pmatrix}$ form a basis for your space. From this, I surmise that this is a question coming immediately after you have covered bases in a linear algebra class or text (which is fine). Does this make sense? It is important to understand bases in a linear algebra class - they form the basis for many things.
A: An arbitrary element of $W$ is 
$$
A=\begin{pmatrix} a & a \\a & b\end{pmatrix}=a\begin{pmatrix} 1 & 1 \\1 & 0\end{pmatrix}+b\begin{pmatrix} 0 & 0 \\0 & 1\end{pmatrix}
$$
So a basis of $W$ is $\{\begin{pmatrix} 1 & 1 \\1 & 0\end{pmatrix}, \begin{pmatrix} 0 & 0 \\0 & 1\end{pmatrix}\}$
