Exercise 20

The fact that for any linear map the rank plus the nullity equals the dimension of the domain shows that a necessary condition for the existence of a homomorphism between two spaces, onto the second space, is that there be no gain in dimension. That is, where $h\colon V\to W$ is onto, the dimension of $W$ must be less than or equal to the dimension of $V$.

• Show that this (strong) converse holds: no gain in dimension implies that there is a homomorphism and, further, any matrix with the correct size and correct rank represents such a map.

• Are there bases for $\mathbb{R}^{3}$ such that this matrix

  $$H=\begin{pmatrix}1&0&0\\ 2&0&0\\ 0&1&0\end{pmatrix}$$

  represents a map from $\mathbb{R}^{3}$ to $\mathbb{R}^{3}$ whose range is the $xy$ plane subspace of $\mathbb{R}^{3}$?