# What is the dimension of the space of traceless $n × n$ matrices?

I am trying to figure out what would be the dimenstion of the space of the traceless $n \times n$ matrices. Please help me with this question.

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This question does not show any research effort – The Chaz 2.0 Sep 23 '11 at 4:45

The space of $n\times n$ matrices has dimension $n^2$. Now we know that the trace of a matrix is the sum of the elements on its diagonal. So elements off diagonal have no effect on a matrix's trace. Hence are we free to choose such elements?
There are $n$ elements on the diagonal and say we choose $n-1$ of them and stop at the entry $a_{nn}$. Is there a constraint on how I choose $a_{nn}$?
Alternatively, note that $\operatorname{tr} : M_n(k) \to k$ is a linear operator on the space of $n \times n$ matrices over the field $k$... – Zhen Lin Sep 23 '11 at 7:21