Basis for space of matrices in $\mathbb M_2(\mathbb R)$ Given that $G=\left\{ \left(\begin{array}{cc}
a & -a\\
b & c
\end{array}\right):a,b,c\in\mathbb{R}\right\} $ and $H=\left\{ \left(\begin{array}{cc}
x & y\\
z & -z
\end{array}\right):x,y,z\in R\right\} $ are matrices in  $\mathbb M_2(\mathbb{R)}$.
(1) Find a basis for $G$. 
(2) Find a basis for $H$ 
(3) Find a basis for $G\cap H$
For the first two of these could be argue that since both of the matrices 
are subsets of $\mathbb M_2(\mathbb R)$that a basis for each of the
matrices would be the standard basis for 
$\mathbb M_2(\mathbb R)$ ie $\left\{ \left(\begin{array}{cc}
1 & 0\\
0 & 0
\end{array}\right)\left(\begin{array}{cc}
0 & 1\\
0 & 0
\end{array}\right)\left(\begin{array}{cc}
0 & 0\\
1 & 0
\end{array}\right)\left(\begin{array}{cc}
0 & 0\\
0 & 1
\end{array}\right)\right\} $?. 
For the intersection of $G$ and $H$ 
I am unsure what a basis would be ? 
 A: Hint: How many independent parameters do you have for each set?
I'll complete it if you need more help.
First edit:
For example, for $G$, you can write each element in $G$ like this $$ \left( \begin{array}{cc}
a & -a\\
b & c
\end{array}\right) = a \left( \begin{array}{cc}
1 & -1\\
0 & 0
\end{array}\right)+b \left( \begin{array}{cc}
0 & 0\\
1 & 0
\end{array}\right)+ c \left( \begin{array}{cc}
0 & 0\\
0 & 1
\end{array}\right)$$
Therefore, the set $\{\left( \begin{array}{cc}
1 & -1\\
0 & 0
\end{array}\right),\left( \begin{array}{cc}
0 & 0\\
1 & 0
\end{array}\right),\left( \begin{array}{cc}
0 & 0\\
0 & 1
\end{array}\right)  \}$ spans $G$. You can show that these matrices are also linearly independent, it's easy to see that. Therefore they form a basis for $G$. 
Can you do the same for $H$ and find a basis for it?
Can you see what kind of matrices will lie in $H \cap G$?
Second edit:
Well, if something lies in both $G$ and $H$, then it must have the forms of elements in both sets.
So, if $A$ is in $G \cap H$ we can find $a,b,c,x,y,z \in \mathbb{R}$ such that:
$$A= \left( \begin{array}{cc}
a & -a\\
b & c
\end{array}\right)=\left( \begin{array}{cc}
x & y\\
z & -z
\end{array}\right)$$
Since the two matrices are equal, we realize that $A$ must look like this:
$$A=\left( \begin{array}{cc}
x & -x\\
z & -z
\end{array}\right)$$
Therefore:
$$G \cap H = \{ \left( \begin{array}{cc}
x & -x\\
z & -z
\end{array}\right): x,z \in \mathbb{R} \}$$
Can you find a basis for $G \cap H$ now?
