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

I will not answer your question directly for you have not put any effort into it. However here are some questions that you can ponder about that may or may not help.

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 $a_{nn}$?

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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