I would like to know all subspaces of a $2\times 2$
diagonal matrix. This is a problem from a book, Introduction to Linear Algebra by Dr. Strang. In the problem, I would like to find all subspaces of a $2\times 2$ diagonal matrix.

The answer says that

> 
> "The subspaces of D are D itself, the zero matrix by itself, and all
> the “one-dimensional” subspaces that contain all multiples of one
> fixed matrix."
> 
> 
> 

the fixed matrix is \begin{bmatrix} d1 & 0\\ 0 & d2 \end{bmatrix}

I understand that the subspaces contain D itself and the zero matrix. However, what does it mean that “one-dimensional” subspaces that contain all multiples of one
fixed matrix" ? The related question is posted in [here](https://math.stackexchange.com/questions/1450033/subspaces-of-the-space-of-all-2-times-2-diagonal-matrices), but I do not really understand it.