Given that the $V$ is the set of all real $2\times2$ matrices, how would one justify that $W$ is a subset of $V$, if
$$W = \left\{ {\begin{pmatrix} a & b\\ -b & a \end{pmatrix}\bigg|\ a,b\in\mathbb{R}}\right\}$$
My approach uses 3 criteria i), ii) and iii):
i) $0\in W$
I let $a,b=0$, to show that $0 = {\begin{pmatrix} 0 & 0\\ 0 & 0 \end{pmatrix}\in W}$
ii) If $v\in W$ and $w\in W$, then $v+w \in W$
I let $v = {\begin{pmatrix} 1 & 2\\ -2 & 1 \end{pmatrix}\in W}$ and $v = {\begin{pmatrix} 3 & 4\\ -4 & 3 \end{pmatrix}\in W}$ to show that $v+w = \begin{pmatrix} 4 & 6\\ -6 & 4 \end{pmatrix} \in \mathbb{R}$
iii) If $v\in W$ and $k\in \mathbb{R}$, then $kv\in W$, where $k$ is a scalar constant
I let $v = \begin{pmatrix} 1 & 2\\ -2 & 1 \end{pmatrix}$ and $k = 2$, so, $ kv = 2\begin{pmatrix} 1 & 2\\ -2 & 1 \end{pmatrix} = \begin{pmatrix} 2 & 4\\ -4 & 2 \end{pmatrix}\in W$
I would like to know if I've correctly justified that each of the 3 criteria are valid, and that $W$ is indeed a subspace of $V$? If not, what is the correct way of doing this?
The idea is correct but to be more general and rigorous we should use generic elements for W, as for example for property ii)
$$v_1 = {\begin{pmatrix} a_1 & b_1\\ -b_1 & a_1 \end{pmatrix}\in W}\quad v_2 = {\begin{pmatrix} a_2 & b_2\\ -b_2 & a_2 \end{pmatrix}\in W}\\\implies v_1 +v_2= {\begin{pmatrix} a_1+a_2 & b_1+b_2\\ -b_1-b_2 & a_1+a_2 \end{pmatrix}\in W}$$
and similarly for property iii).