Prove that if $X$ and $Y$ are in $\U0001d52c(n)$ then $[X,Y]$ is also in $\U0001d52c(n)$ .

# Week 2

## Session 1

### Pre-class videos

### Pre-class exercises

Exercise:

Exercise:

Show that $X=\left(\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)$ and $Y=\left(\begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{array}\right)$ are in $\U0001d530\U0001d529(2,\mathbf{C})$ .

## Session 2

### Pre-class videos

### Pre-class exercises

Exercise:

Check that the Jacobi identity holds for the commutator bracket $[X,Y]:=XY-YX$ .

Exercise:

Why is the tangent vector to $\mathrm{exp}(sX)$ at $s=0$ equal to $X$ ?

## Additional