I'm working with the set of trace zero matrices, $\mathfrak{sl}(V)\subseteq\mathfrak{gl}(V)$ of endomorphisms of a vector space $V$.
The problem asks us to represent $ad_x, ad_y, ad_h$ in terms of the basis elements
$x = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$
$y = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$
$h = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$
I have computed that
$ad_x(y)=h$,
$ad_x(h)=-2x$,
$ad_x(x)=0$,
Similarly,
$ad_y(y)=0$,
$ad_y(x)=-h$,
$ad_y(h)=2y$
finally,
$ad_h(h)=0$,
$ad_h(x)=2x$,
$ad_h(y)=-2y$.
This is my first course in Lie algebras, so I'm kind of stuck. In linear algebra, if I could show how a transformation acted on a basis, it'd be easy to write the result in terms of the basis vectors and write down the linear transformation in a matrix. I computed the adjoints and represented them in terms of the basis matrices, but I'm stuck now. If someone could give me a hint, or show how to do it for one of the adjoints, I'd be okay from there. Thanks