Worksheet 12

1. Find a basis of the null space of the matrix \(A = \begin{pmatrix} 0 & -1 & 2 & 1 \\ 4 & 3 & -2 & 0 \\ 1 & 0 & 1 & 1 \end{pmatrix}\) What is the dimension?

2. Find a basis of the range of the matrix \(A = \begin{pmatrix} 1 & -1 & 0 \\ 3 & 1 & 4 \\ -1 & 1 & 0 \\ 5 & -1 & 4 \end{pmatrix}\) What is the dimension?

3. Suppose $V$ is a vector space of dimension $d$. Show that that there is a chain of subspaces \(0 \subsetneq U_1 \subsetneq U_2 \subsetneq \cdots \subsetneq U_{d-1} \subsetneq V\) with $\dim U_i = i$.