(1) Prove that any linear transformation T : 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....

Extracted text: (1) Prove that any linear transformation T : 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?