Exercise 11

Take each matrix to represent $h\colon\mathbb{R}^{m}\to\mathbb{R}^{n}$ with respect to the standard bases. For each (i) state $m$ and $n$. Then set up an augmented matrix with the given matrix on the left and a vector representing a range space element on the right (e.g., if the codomain is $\mathbb{R}^{3}$ then in the right-hand column put the three entries $a$, $b$, and $c$). Perform Gauss-Jordan reduction. Use that to (ii) find $\mathscr{R}(h)$ and $\operatorname{rank}(h)$ (and state whether the underlying map is onto), and (iii) find $\mathscr{N}(h)$ and $\operatorname{nullity}(h)$ (and state whether the underlying map is one-to-one).

• $\begin{pmatrix}2&1\\ -1&3\end{pmatrix}$

• $\begin{pmatrix}0&1&3\\ 2&3&4\\ -2&-1&2\end{pmatrix}$

• $\begin{pmatrix}1&1\\ 2&1\\ 3&1\end{pmatrix}$