Exercise 5

Show that linear maps preserve the linear structures of a space.

• Show that for any linear map from $\mathbb{R}^{n}$ to $\mathbb{R}^{m}$, the image of any line is a line. The image may be a degenerate line, that is, a single point.

• Show that the image of any linear surface is a linear surface. This generalizes the result that under a linear map the image of a subspace is a subspace.

• Linear maps preserve other linear ideas. Show that linear maps preserve “betweeness”: if the point $B$ is between $A$ and $C$ then the image of $B$ is between the image of $A$ and the image of $C$.