Answer Happy

T: M2,2 A (a) Show that T is a linear transformation. → M2,2 A
(b) Find the matrix representation [T]s of T using the standard ordered basis for M2,2, (see p.200, Topic 4.5 of the Lecture Notes.)
(c) Find the image, kernel, rank and nullity of T.
(d) Is T invertible? Why or why not?
Standard Bases The standard basis for R" is {(1,0,0,...,0), (0, 1, 0, & 0), (0,0,0....,0,1)). en These vectors are typically denoted as (e₁,e₂,en). (In R³ the standard basis is also denoted by (i.j, k}) kz The dimension of R" is n. mxn real matrices. The standard basis for Mm.n is 07 To 1 *** 1 0 00 0 0 0 0 00 ... (6 }} 1 00 0 --X The dimension of Mm.n is mp e. g. M₂, 2 = basis = { [68] [%], [8], []}= dini real polyromial of degree in The standard basis for Pn is P(x)=ay²H+a+n. Anx {1, x,x²,...,x"}. give typical polynomial in The dimension of P is n + 1. eig P₂ hrs basis {1₁x₁x²} = dim P₂=3 Let P be the set of all polynomials of all possible degrees: P = {ao + ax ++ anx" | ne N, ap. ₁,..., an ER} where N denotes the set of natural numbers. P is an infinite dimensional vector space, since for each integer k> 1, we can find a linearly independent set {1,...,x} with more than k elements. The standard basis for Pis