Subspaces of the space of all $2\times 2$ diagonal matrices The exercise is to describe all the subspaces of $D$, the space of all $2\times 2$ diagonal matrices.
I just would have said $I$ and $Z$ initially, since you can't do much more to simplify a diagonal matrix. 
The answer given is:

The subspaces of $\mathbb{R}^4$ are $\mathbb{R}^4$ itself, three-dimensional planes $\mathbf{n}\cdot\mathbf{v}=0$, two-dimensional subspaces ($\mathbf{n}_1\cdot\mathbf{v}=0$ and $\mathbf{n}_1\cdot\mathbf{v}=0$), one-dimensional lines through $(0,0,0,0)$, and $(0,0,0,0)$ by itself.

I cannot understand how $D$ is $\mathbb{R}^4$, much less understand the rest of the answer.
I can see how we might take the columns of $D$ and form linear combinations from them, but those column vectors are in $\mathbb{R}^2$.
 A: When we view the set of $2\times 2$ matrices as a (vector) space, we have to make clear how we do that, i.e., what it measn to add two elements of this space and what it means to multiply an element of this space by a real number. The somewhat obvious choices for these operations are
$$\begin{pmatrix}a&b\\c&d\end{pmatrix} +\begin{pmatrix}a'&b'\\c'&d'\end{pmatrix}=\begin{pmatrix}a+a'&b+b'\\c+c'&d+d'\end{pmatrix}$$
and 
$$\lambda \cdot \begin{pmatrix}a&b\\c&d\end{pmatrix} =\begin{pmatrix}\lambda a&\lambda b\\\lambda c&\lambda d\end{pmatrix}.$$
Then one notices that the map $$(a,b,c,d)\mapsto \begin{pmatrix}a&b\\c&d\end{pmatrix} $$
is (again, somewhat obviously) an isomorphism of $\mathbb R^4$ with the space f $2\times 2$ matrices. Therefore the classification of subspaces of the space of $2\times 2$ matrices is "the same" as the classification of subspaces of $\mathbb R^4$. 
For this task it does not matter that $2\times 2$ matrices have an additional structure beyond that as a vector space (namely, that matrices can be multiplied).
Morover, the space of diagonal $2\times 2$ matrices is a subspace of the space of all $2\times 2$ matrices that is isomorphic to $\mathbb R^2$. Again, an explicit isomorphism is straightforward to write down: $(x,y)\mapsto \begin{pmatrix}x&0\\0&y\end{pmatrix}$.
