This is a linear algebra problem

I want to know how to solve these problems because the professor said there is no answer. These problems are the review questions for midterm.

Could you solve all these problems?

I do not know how to solve these problems, so I need explanation.

This is my last chance to revise question.

Thank you

10. Can you generalize the same procedure. for higher order polynomials? How would the matrices [T] 3 and [T]g look like then? Which one would to prefer and why? A matrix A E MnxnR) is called idempotent if A2 = A - A = A. (a) Show that if A is idempotent then R(A) N(A) = {0} Let V and W be vector spaces, and let S be a subset of V. Dene 5'0 = {T E (V,W) : T(m) = 0 for all a: E S}. Prove the following statements. (a) 5'0 is a subspace of (V,W) (b) If 31 and 5'2 are subsets of V and 81 Q 5'2, then 38 Q S? (c) If V1 and V2 are subspaces of V, then (V1 U V2)0 = (Vl + V2)0 = V? F] V3 Let 3 = {121,122, . . . ,vn} be a basis for a vector space V and T : V > V be a linear transfor- mation. Prove that [TM is upper triangular if and only if T0055) 6 span ({v1,v2, . . . ,vj}) for j = 1,2,...,n