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I am trying to find the tangent space of SL(n,real) where A(0) is defined to be the identity matrix.

First of all I worked on the case when n=2 and found that the tangent space was

[tex]A = \left( \begin{array}{ccc}

a & b \\

c & -a

\end{array} \right) [/tex]

where a,b,c belong to the reals,

so I made the conjecture that for n in general, the tangent space would be the space of traceless matrices.

I attempted to prove this by showing that the tangent space and the space of traceless matrices were subsets of each other. Whilst I could show that an arbitary element of the tangent space is traceless, I could not show the converse.

Do I just need to try harder or is my conjecture just plain wrong?

PS. I used the standard result: d/dt (detA(0)) = tr(dA(0)/dt)

I have reason to believe that det(exp(A)) = exp(tr(A)) may also be important but have not found a way of using this yet.

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# Homework Help: Vector Calculus

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