A projection is a nonzero linear operator P such that P2-P. Let v be an eigenvector with eigenval...
5. Let A, B E Mmxm(R) and let v be an eigenvector of A with eigenvalue 1, and v be an eigenvector of B with eigenvalue j. (a) Show that v is an eigenvector of AB. What is the corresponding eigenvalue? (b) Show that v is an eigenvector of A+B. What is the corresponding eigenvalue?
4. (a) (6 marks) Let A be a square matrix with eigenvector v, and corresponding eigenvalue 1. Let c be a scalar. Show that A-ch has eigenvector v, and corresponding eigenvalue X-c. (b) (8 marks) Let A = (33) i. Find the eigenvalues of A. ii. For one of the eigenvalues you have found, calculate the corresponding eigenvector. iii. Make use of part (a) to determine an eigenvalue and a corresponding eigenvector 2 2 of 5 - 1
5. Let T: P2 Dasis for P2. P2 be the linear operator defined as T(P(x)) = p(5x), and let B = {1,x, x2} be the standard Find [T]b, the matrix for T relative to B. Let p(x) = x + 6x2. Determine [p(x)]B, then find T(p(x)) using [T]s from part a. Check your answer to part b by evaluating T(x + 6x2) directly.
2. (10 points) Suppose v is an eigenvector of A with eigenvalue X, and let c be a real number. Show that v is an eigenvector of A+cI, where I is the appropriately sized identity matrix. What is the corresponding eigenvalue?
Let T be a linear operator on F2. Prove that if v f 0 is not an eigenvector for T, then v is a cyclic vector for T. Conclude that either T has a cyclic vector T is a scalar multiple of the identity.
4. Let TV - V be a linear operator on a finite dimensional inner product space V and P be the orthogonal projection of V onto the subspace W of V. a) Show that is invariant under T if and only if PTP = TP. b) Show that w and we are both invariant under 7 If and only if PT = TP
(10](3) State the definition of eigenvalue. It begins: Dn: (eigenvalue) Let 7V V be a linear operator and 1 € R. A is an eigenvalue of TW [10(4) 13 5 5 GIVEN: A E M(3,1), A = -2 -1 -2 1 2 -1 0 the linear operator, T:M(3,1) - M(3,1), Tz = At and v = -1 EM(3,1) and v is an eigenvector of T. FIND: The eigenvalue, 1, of T associated with u.
Find the eigenvalues for A and any one (nonzero) eigenvector 11. (7 pts) Let A = A. Show all work
3. EIGENVALUE AND EIGENVECTOR BASICS Select at most one of the following problems. You may use the rest of this page and the back of this sheet NOTE: No points will be earned if you do not justify your statements. (1) (10 points) A projection operator T is one that satisfies the equation (2) (15 points) Discuss the validity of the following statement. If x and y T2 -T. What are the possible eigenvalues of T? are eigenvectors of a...
4. Let G : P(R) → P2(R) be a linear map given by G(u)(x) = (x + 1)u'r) - ur). Is G diagonalizable? If it is, find a basis of P(R) in which G is represented by a diagonal matrix 5. Let V = P2(C). Show that the operator (.) given by (u, v) = u(0) v(0) + u(1) v(1) + u(2) v(2) Vu, v E V is an inner product on V.