Hey, I was wondering if someone could help me find a basis for the space of all nxn matricies, over R, with trace 0.

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- September 29th 2010, 07:39 PMHelloWorld2Basis for matrices of trace 0
Hey, I was wondering if someone could help me find a basis for the space of all nxn matricies, over R, with trace 0.

- September 29th 2010, 07:53 PMMacstersUndead
It's been a long while, but maybe I can give a small push. Take this with a grain of salt

We know that

If then all .

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

and

EDIT:// ** with - September 29th 2010, 08:52 PMHelloWorld2
But if the 1,1th entry is 1, and the 2,2th entry is -1, we also have a zero trace.

- September 29th 2010, 08:55 PMMacstersUndead
- September 30th 2010, 06:29 AMHallsofIvy
A basis consists of all n by n matrices in which

1) For i and j from 1 to n with , with entries and all other entries 0 and

2) For i and j from 1 to n with , with entries and .

and all other entries 0.

If n= 2, those would be , , and .