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)...
Problem 4. Give an example of a linear operator T on a finite-dimensional vector space such that T is not nilpotent, but zero is the only eigenvalue of T. Characterize all such operators. Problem 5. Let A be an n × n matrix whose characteristic polynomial splits, γ be a cycle of generalized eigenvectors corresponding to an eigenvalue λ, and W be the subspace spanned by γ. Define γ′ to be the ordered set obtained from γ by reversing the...
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...
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...
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
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*...
Problem 6. Let V, W, and U be finite-dimensional vector spaces, and let T : V → W and S : W → U be linear transformations (a) Prove that if B-(Un . . . , v. . . . ,6) is a basis of V such that Bo-(Un .. . ,%) s a basis of ker(T) then (T(Fk+), , T(n)) is a basis of im(T) (b) Prove that if (w!, . . . ,u-, υ, . . . ,i)...
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...
Let V be an inner product space and u, w be fixed vectors in V . Show that T v = <v, u>w defines a linear operator in V . Show that T has an adjoint, and describe T ∗ explicitly
4. Let T be the linear operator on F which is represented in the standard ordered basis by the matrix c0 0 01 Let W be the nll space of T - c/. (a) Prove that W is the subspace spanned by 4 (b) Find the monic generators of the ideals S(u;W), S(q;W), s(G;W), 1 4. Let T be the linear operator on F which is represented in the standard ordered basis by the matrix c0 0 01 Let W...