Basis for Special linear group

Question

I'm looking for basis for the tangent space $T_eSL_2(R)$ of the real special linear group $SL_2(R)$. I only know that group have $X$ matrices with $Trace(X)=0$. How can I do that? Any hint? Thank you.

Answer 1

The tangent space $T_e SL_2\Bbb R \cong \mathfrak{sl}_2$ which is the following algebra over $\Bbb R$:

$$\{ X\in M_2\Bbb R: \text{tr}X=0\}$$

Now it easy to see that a basis is given by

\begin{pmatrix} 0&1 \\ 0&0 \end{pmatrix}

\begin{pmatrix} 0&0\\ 1&0 \end{pmatrix}

\begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}