Worksheet 10

1. Which of the following are subspaces and which are not:
   • Upper triangular matrices in $\operatorname{Mat}_{m,n}(k)$.
   • Invertible matrices in $\operatorname{Mat}_{n,n}(k)$.
   • Diagonal matrices in $\operatorname{Mat}_{n,n}(k)$.
   • Square matrices satisfying $A^r = I$ for some $r$.

2. Find a spanning set for the null space of the matrix \(A = \begin{pmatrix} 1 & -1 & 0 & 1 \\ 0 & 2 & 2 & -1 \\ 1 & 1 & 2 & 0 \end{pmatrix}\)

3. Suppose we have two vectors spaces $V_1$ and $V_2$ with two subset $U_1 \subseteq V_1$ and $U_2 \subseteq V_2$. Is the following true? \(\operatorname{Span}(U_1) \times \operatorname{Span}(U_2) = \operatorname{Span}(U_1 \times U_2)\)