1. Let T be the matrix T=10 3 acting on the complex vector space V C3 (a) Recall how T defines the structure of a C-mod...
Problem 4 Let V be the vector space of functions of the form f(x) = e-xp(x), where p(x) is a polynomial of degree (a) Find the matrix of the derivative operator D = d/dx : V → V in the basis ek = e-xXk/k!, k = 0, 1, . .. , n, of V. (b) Find the characteristic polynomial of D. (c) Find the minimal polynomial of D n. Problem 4 Let V be the vector space of functions of...
Problem 4 Let V be the vector space of functions of the form f(x) = e-xp(x), where p(x) is a polynomial of degree (a) Find the matrix of the derivative operator D = d/dx : V → V in the basis ek = e-xXk/k!, k = 0, 1, . .. , n, of V. (b) Find the characteristic polynomial of D. (c) Find the minimal polynomial of D n.
Let V be a finite-dimensional vector space over C and T in L(V). Prove that the set of zeros of the minimal polynomial of T is exactly the same as the set of the eigenvalues of T.
Let V be a finite-dimensional vector space and let T L(V) be an operator. In this problem you show that there is a nonzero polynomial such that p(T) = 0. (a) What is 0 in this context? A polynomial? A linear map? An element of V? (b) Define by . Prove that is a linear map. (c) Prove that if where V is infinite-dimensional and W is finite-dimensional, then S cannot be injective. (d) Use the preceding parts to prove...
4. Let T be a linear operator on the finite-dimensional space V with eharacteristie polynomial and minimal polynomial Let W be the null space of (T c) Elementary Canonical Forms Chap. 6 226 (a) Prove that W, is the set of all vector8 α in V such that (T-cd)"a-0 for some positive integer 'n (which may depend upon α). (b) Prove that the dimension of W, is di. (Hint: If T, is the operator induced on Wi by T, then...
Let A be an invertible linear operator on a finite-dimensional complex vector space V. Recall that we have shown in class that in this case, there exists a unique unitary operator U such that A=UA. The point of this exercise is to prove the following result: an invertible operator A is normal if and only if U|A= |AU. a) Show that if UA = |A|U, then AA* = A*A. Now, we want to show the other direction, i.e. if AA*...
Let T be a linear operator on a finite dimensional vector space with a matrix representation A = 1 1 0 0] 16 3 2 1-3 -1 0 a. (3 pts) Find the characteristic polynomial for A. b. (3 pts) Find the eigenvalues of A. C. (2 pts) Find the dimension of each eigenspace of A. d. (2 pts) Using part (c), explain why the operator T is diagonalizable. e. (3 pts) Find a matrix P and diagonal matrix D...
A1. Let (A, B, C, D) be a SISO system in which A is a (n x n) complex matrix and B a (n x 1) column vector, let -1 V = {£ajA*B: aj e C; j= 0, ...,n- (i) Show that V is a complex vector space. (ii) Show that V has dimension one, if and only if B is an eigenvector of A AX for X E V. Show that S defines a linear map from S: V...
Suppose that V is a 3-dimensional vector space over a field F and T : V → V is a linear tion such that the corresponding F[x]-module structure on V is given by 7. V F[x]/(x3-x2-x + 1). Among the matrices A, B, and C given below, which are the matrix of T in some basis for V. Explain 1 1 0 0 0-1 B-10 1 A 0 1 0 0 1 1 0 0 -1 0 0 -1 (Note:...
3. Let t be the co-ordinate on A (C) and let z, y be the co-ordinates on A2(C). Let f 4z? + 6xy + x-2y® E C[x, y] and let C be the curve C-V((f)) C A2(C) (You may assume without proof that f is an irreducible polynomial, therefore C is irreducible and I(C)- (f).) (a) Show that yo(t) = (2t3, 2t2 + t) defines a morphism p : A1 (C) → C. [3 marks] (b) Show that (z. У)...