Hey, I was wondering if someone could help me find a basis for the space of all nxn matricies, over R, with trace 0.
It's been a long while, but maybe I can give a small push. Take this with a grain of salt
We know that
$\displaystyle \mathrm{tr}(A) = a_{11} + a_{22} + \dots + a_{nn}=\sum_{i=1}^{n} a_{i i}$
If $\displaystyle \mathrm{tr}(A) = 0$ then all $\displaystyle a_{i i} = 0$.
A basis for the space of all nxn matricies, then, is the set of all matricies with 1 as an ij-th entry **, and all the rest being 0. go through all the possibilities.
ex. for a 2x2 matrix, the basis would be
$\displaystyle
\left[ {\begin{array}{cc}
0 & 1 \\
0 & 0 \\
\end{array} } \right]
$ and $\displaystyle
\left[ {\begin{array}{cc}
0 & 0 \\
1 & 0 \\
\end{array} } \right]
$
EDIT:// ** with $\displaystyle i \neq j$
A basis consists of all n by n matrices in which
1) For i and j from 1 to n with $\displaystyle i\ne j$, $\displaystyle A_{ij}$ with entries $\displaystyle a_{ij}= 1$ and all other entries 0 and
2) For i and j from 1 to n with $\displaystyle j> i$, $\displaystyle B_{ij}$ with entries $\displaystyle b_{ii}= 1$ and $\displaystyle b_{jj}= -1$.
and all other entries 0.
If n= 2, those would be $\displaystyle B_{12}= \begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix}$, $\displaystyle A_{12}= \begin{bmatrix}0 & 1 \\ 0 & 0\end{bmatrix}$, and $\displaystyle A_{21}= \begin{bmatrix}0 & 0 \\ 1 & 0\end{bmatrix}$.