1)Is $$ \Bigl\{ \begin{pmatrix} x & x+y \\ 0 & y \end{pmatrix} \Bigm| x,y\in\mathbb R \Bigr\} $$ a subspace of the real vector space of $2\times 2$ matrices? If so, show it. If not, why not?
2) Is $$ \Bigl\{ \begin{pmatrix} x & x+y \\ 0 & y \end{pmatrix} \Bigm| x,y\in\mathbb Q \Bigr\} $$ a subspace of the real vector space of $2\times 2$ matrices? If so, show it. If not, why not?
I think that only the first one is true and I am not sure about the second one, but I have no clue on how to prove it.
What I think is the first step is that $1(x+y)-1(x)-(y) = 0$, and because $c=0$, it forms a subspace.
For the first one:
You could easily notice that the zero matrix is in this set (just place $x=0$ and $y=0$). Then if you have two matrix of the set you can easily find that also the sum is part of the set:
$$\begin{pmatrix} x & x+y \\ 0 & y \end{pmatrix} + \begin{pmatrix} x' & x'+y' \\ 0 & y' \end{pmatrix} = \begin{pmatrix} x + x' & (x+x') + (y+y') \\ 0 & y+y' \end{pmatrix}$$
with $x+x', y+y' \in \ \Bbb R$
With the same reasoning you can find that your set is closed for the scalar multiplication. Let $\lambda \in \ \Bbb R$:
$$\lambda\begin{pmatrix} x & x+y \\ 0 & y \end{pmatrix} = \begin{pmatrix} \lambda x & \lambda(x+y)\\ 0 & \lambda y \end{pmatrix}$$
with $\lambda x,\lambda y \in \ \Bbb R$. So your first set is a vectorial subspace.
The second exercise is pretty much identical and I'm sure that now you can solve it by yourself.