Linear algebra-vector space and mapping

Question

Question:

Linear Algebra, Vector Space and Mapping

Let β = {x₁, ..., x_n} be a basis for a vector space V, and let P be the mapping P(a₁)(x₁) + ... + (a_n)(x_n)) = (a₁)(x₁) + ... + (a_k)(x_k).

a) Show that Ker(P) = Span({x_k+1, ..., x_n}) and Im(P) = Span ({x₁, ..., x_k})

b) Show that P^2 = P

c) Show conversely that if P:V→V is any linear mapping such that P^2 = P, then there exists a basis Beta for V such that P takes the form given in part a. (Hint: Show that P^2 = P implies that V = Ker(P) + Im(P). These mappings are called projections. The orthogonal projections we studied in Chapter 4 are special cases.