Let us consider the inner product space R" with its standard inner product. Let T: R"R" be defined by T(,2,)=(x - 1, 23-, 21 In)- Give an explicit expression for the adjoint T*. Suppose 8 M P(R) is a linear map whose matrix representation with respect to the bases is S= -{[] [] [] []} 01 Use this to compute 3 2 1 2 2 2 10 -2-100 2 1 1 2 and {x+x+1, 2+1,1} [9] a. Show that a 3 x 3 matrix A = [A, A, A] is invertible if = -4 b. Assume that T: R R be a transformation given by T(x,y) = (1-xy.x+y). Check if T is a linear transformation. Give a counter example. An x n matrix. A has a set of n linearly independent eigenvectors. Prove that A is diagonalisable. Given H = 2 [10 Marks] 0 21 Determine the characteristic polynomal of matrix H by employing the algorithm for the determination of the eigenvalues of standard Hessenberg matrix.