Question

Prove that any linear transformation ? : R? → R? maps a line passing through the origin to either the zero vector or a line passing through the origin. Generalize this for planes and hyperplanes. What are the images of these under linear transformations?

Answer 1

Consider a line passing through the origin in . It can be written in the form:

Then, applying on , we have:

If , then which is the zero vector. Else , which is a line passing through the origin.

Generalizing is straightforward:

Any linear transfomation that is an automorphism maps subspaces to subspaces of same or less dimension or the zero vector.

Then consider the specific subspaces of dimension 2 (planes) or that of codimension 1 (hyperplanes).

The images are respectively the subspaces they are mapped to. For the plane, the image is either a plane through the origin, or a line through the origin or the zero vector.

For the hyperplane, the possible images are subspaces with codimension more than 1 and less than n.