(5) Let V be a finite dimensional vector space with basis {v1, ... , Un} and T E L(V) Show the following are equivalent: (a) The matrix for T is upper triangular. (b) T(vk) E Span(v1,.., vk) for all k. (c) Span(v1, ... , Vk) is T-invariant for all k.

Consider the vectors u=(6,2,4) and v=(1,2,0) from Example 10. Without using Theorem 5.9, show that among all the scalar multiples cv of the vector v, the projection of u onto v is the closest to u that is, show that d(u,projvu) is a minimum.