Is this $2 \times 2$ matrix a subspace

Question

I have the following vector space here with a subset of $W$ of $V$:

$V=M_{2 \times 2}(\mathbb{R}),W=\begin{bmatrix}x & y\\ 0 & -x+2y\\\end{bmatrix}|x,y \in \mathbb{R}$

I have to show if this is a subspace or not.

$1)$ The $0$ condition is safisfied obviously if $x=y=0$.

$\begin{bmatrix}0 & 0\\ 0 & -(0)+2(0)\\\end{bmatrix}|x,y \in \mathbb{R}$

$\begin{bmatrix}0 & 0\\ 0 & 0\\\end{bmatrix}|x,y \in \mathbb{R}$

$2)$ Multiplication by a scalar is defined of course.

$\begin{bmatrix}cx & cy\\ 0 & -cx+2cy\\\end{bmatrix}$

$\begin{bmatrix}x_1 & y_1\\ 0 & -x_1+2y_1\\\end{bmatrix}$

For some choice of scalar $c$ in $\mathbb{R}$.

Closure under addition is something I am not sure about. If I add an arbitrary term to the $0$ in the matrix, I don't get $0$ as it was defined in the original matrix. Does that mean I don't have a subspace? I need some clarification on this. Thanks!

Answer 1

$W$ is a subspace. Note that $\begin{bmatrix}x & y\\ 0 & -x+2y\\\end{bmatrix} +\begin{bmatrix}x' & y'\\ 0 & -x'+2y'\\\end{bmatrix}=\begin{bmatrix}u & v\\ 0 & -u+2v\\\end{bmatrix}$ where $u=x+x'$ and $v=y+y'$.