basis of space of all $3\times3$ matrices and basis of space of all $3\times3$ matrices with rank $0, 1$ or $2$

Question

I have a set of all $3\times3$ matrices $B$ that have rank $0, 1$ or $2$ - what is the basis of a subspace that this set generates?

I came to the conclusion that this subspace consists of lines, planes and the null vector and it generates the whole space of $\Bbb R^3$, but I don't know what is the basis of this subspace - is it just the set of all the $B$'s?

EDIT: my original thinking was very wrong, the questions I ended up with are what is the basis of the space of all $3\times3$ matrices and what is the basis of the subspace - all $3\times3$ matrices with rank $0, 1$ or $2$

Answer 1

The vector space of all $3 x 3$ matrices is not $R^3$. You can verify that the space has dimension $9$ because you will need $9$ vectors for a basis. Probably the most likely ones would be $\begin{bmatrix}1 & 0 & 0\\0 & 0 & 0\\0 & 0 & 0\end{bmatrix}$, $\begin{bmatrix}0 & 1 & 0\\0 & 0 & 0\\0 & 0 & 0\end{bmatrix}$ ... $\begin{bmatrix}0 & 0 & 1\\0 & 0 & 0\\0 & 0 & 0\end{bmatrix}$,$\begin{bmatrix}0 & 0 & 0\\1 & 0 & 0\\0 & 0 & 0\end{bmatrix}$, $\begin{bmatrix}0 & 0 & 0\\0 & 1 & 0\\0 & 0 & 0\end{bmatrix}$, $\begin{bmatrix}0 & 0 & 0\\0 & 0 & 1\\0 & 0 & 0\end{bmatrix}$,$\begin{bmatrix}0 & 0 & 0\\0 & 0 & 0\\1 & 0 & 0\end{bmatrix}$, $\begin{bmatrix}0 & 0 & 0\\0 & 0 & 0\\0 & 1 & 0\end{bmatrix}$, $\begin{bmatrix}0 & 0 & 0\\0 & 0 & 0\\0 & 0 & 1\end{bmatrix}$

Now considering that your subspace only contains matrices with rank $\leq2$ the basis for that space will contain only $6$ of these $9$.

Answer 2

It's unclear which space you're talking about---subspace of what? If you are viewing $3\times3$ matrices as elements of $V=\mathbb R^{3,3}$ i.e. $3\times3$ matrices under addition, then the answer is all of $V$.