please send complete handwritten solution for Q10
Extracted text: 10.(a) Find the standard matrix representation for T: T(x,X2, x; ) = (3x,,x2,x; – x,) (b) If diagonalizable, find the diagonalizing matrix. In other words, after finding the matrix representation of T, solve the eigen-value eigenvector problem for it, thereby producing the diagonalizing matrix composed of the eigenvectors. 11. Prove that similar matrices have the same characteristic polynomial. This next problem is like a "little journey" through some of the higher level concepts we discussed towards the end of our course. Take your time on it! 12.(a) Prove that if v is a fixed vector in a real inner product space V then the mapping T:V →R defined by T(x) = (x,v) is linear. Next ... (b) Let V = R' have the standard inner product and v=[1,0,2] , Compute T (1,1,1). Next... (c) Let V = P, have the inner product (p,q)= Ja,b, +a,b, + a,b, where p(x)= a, +a,x+a,x' and q(x)= b, +b,x+b,x, are any two vectors in V = P, and moreover let v=1+x. Compute T(x+x*). Finally... (d) Let V = P, have the " evaluation" inner product where the evaluations are at the points 1,0,-1, and v=1+x. Compute T(x+x) .