Show that matrix $B$ is a subspace of matrix $V$.

Question

Let's say we have the set $V$ and $B$. I want to show that $B$ is a subspace of $A$ and to do that I showed that you can create $V$ using a combination of variables ($a, b,c $) multiplied with the matrix. Is my reasoning correct?

Answer 1

Let check

$$(2a+b-2c)\begin{bmatrix}0&1\\1&0\end{bmatrix}+a\begin{bmatrix}1&0\\0&1\end{bmatrix}+(c-a)\begin{bmatrix}0&2\\2&1\end{bmatrix}=\\=\begin{bmatrix}a&2a+b-2c+2c-2a\\2a+b-2c+2c-2a&a+c-a\end{bmatrix}=\begin{bmatrix}a&b\\b&c\end{bmatrix}$$

and it is correct.

Note that what we say in that case is that $B$ is a basis for $V$, indeed $V$ has dimension $3$ and the $3$ vectors/matrices of $B$ span $V$.