1. Let V and W be F-vector spaces, and let T: V→ W be a linear transformation. Prove that dim R(T)+dim N(T) = dim V if V is fnite-dimensional.

7. (12 points) Let T: be a linear operator. Show that if T preserves the Euclidqdean distance between any two points, that is, ||T(u) - T(ข)||= ||u- ull for any u, u , then the matrix representation of T relative to the standard basis is an orthogonal matrix.