# Finding a basis for vector space $U$

Let $U$ denote the subspace of $M_{2\times 2}(\mathbb{C})$ defined by

$$U=\left\lbrace\left(\begin{matrix}a&b\\ c&0\end{matrix}\right):a + b + c=0\right\rbrace.$$

How would one find a basis for that vector space? Any clues please.

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Well, the only requirement is that $a+b+c=0$, and since we don't require the matrix to be invertible, a general matrix is then $$\left(\begin{matrix}a&b\\ -a-b&0\end{matrix}\right)$$ Can you work out the basis from here?
Well, the same thing really. First, make one variable dependent on the rest of the variables, so we have $d=-a-b-c$, so work out the general matrix. Then notice that we have 3 "free" variables in $a,b,c$, so the dimension of the basis must be 3. Can you work it out? I'm happy to explain more if you need me to. –  BlackAdder Jul 20 '13 at 10:17