Determine whether the set S (below) is a subspace of $M_2(\mathbb{R})$ (The space f all 2x2 matrices with real entities) $$S = \left\{
\begin{pmatrix}
2a && -b \\
3b && a
\end{pmatrix}\;:\;\; a,b \in \Bbb R\right\}$$
So, I have to find that it is either closed under addition and scalar multiplication, I asked a similar question to this earlier today but made a mistake with how I asked it, so apologies for that. However I am still left with a few difficulties in the proving that is/isn't a subspace of $M_2(\mathbb{R})$ 
Also, can a set S be a subspace of $M_2(\mathbb{R})$ at all? Or does it fail  from the beginning?
 A: Your way to show that S is closed under addition is absolutely correct but it is better to continue like this : 
$\begin{bmatrix}(2x_1+2x_2) & (-y_1-y_2)\\(3y_1+3y_2) & (x_1+x_2)\end{bmatrix} = \begin{bmatrix}2(x_1+x_2) & -(y_1+y_2)\\3(y_1+y_2) & (x_1+x_2)\end{bmatrix}\in S$ since $(x_1+x_2),(y_1+y_2)\in R$ 
I think you can handle multiplication as well.
A: In the comments it seems like you've shown that S is closed under addition, since after you've added the matrices together you get another matrix that belongs to $S$ with $a=x_1+x_2$, $b=y_1+y_2$.  The subtle point is that if $x_1,x_2\in\mathbb{R}$, then their sum is also a real number, and the same for the $y_1,y_2$.  
The only thing left is scalar multiplication, which is to say if $T\in S$ then $c*T\in S$ for all $c\in\mathbb{R}$.
A: The $2\times2$ real matrices form a vector space under matrix addition and multiplication by scalars. Each one of your matrices can be written as
$$
a\begin{pmatrix}2 & 0 \\ 0 & 1\end{pmatrix}+
b\begin{pmatrix}0 & -1 \\ 3 & 0\end{pmatrix}
$$
so $S$ is the set of all linear combinations of the two matrices
$$
\begin{pmatrix}2 & 0 \\ 0 & 1\end{pmatrix},\qquad
\begin{pmatrix}0 & -1 \\ 3 & 0\end{pmatrix}
$$
and as such it is a vector subspace of $M_2(\mathbb{R})$.
