Find a basis for the subspace of M2,2

[Xingconan]

Homework Statement

The Attempt at a Solution

I don't really know how to do this, so I hope someone can give some hints or briefly tell me what I should do.

 

[HallsofIvy]

Well, a good start is to write
\left[\begin{array}{cc}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right]\left[\begin{array}{cc}1 & 0 \\ -1 & -1\end{array}\right]= \left[\begin{array}{cc}-1 & 0 \\ 1 & 1 \end{array}\right]

Multiply them out and get 4 equations relating a₁₁, a₁₂, a₂₁, and a₂₂. Are those equations independent? If not how many "independent" variables do you have (in other words, how many must you know in order to be able to calculate the others?).