19. Show that if T is linear operator then det(T ∗ ) = det(T)
19. Show that if T is linear operator then det(T ∗ ) = det(T) 19. Show...
a. Let be an differential operator. Show that L is a linear operator. b. Let be an differential operator. Show that the kernel of L is a vector space c. Let . Show that the set of functions which satisfy L(u) = g(x,t) form an affine linear subspace. L=(a252 -2) L=(a252 -2) L=(a252 -2) L=(a252 -2) L=(a252 -2) L=(a252 -2)
Suppose V is finite-dimensional, T:V V is a linear operator, and (T-21)(T-31)(T-41) = 0. Show (without resorting to the Cayley-Hamilton theorem) that if is an eigenvalue of T, then = 2, 3, or 4. Suggestion: compute (T – 21)(T – 31)(T – 41)Ū, where ū is an eigen vector of Twith eigenvalue .
Suppose T: V V is a linear operator. Suppose p(x) = (1-r)(x- s) has distinct real roots (rs) and that and p(T) is the zero operator. Show that V is spanned by eigenvectors of T with eigenvalues r and s. Suppose T: V V is a linear operator. Suppose p(x) = (1-r)(x- s) has distinct real roots (rs) and that and p(T) is the zero operator. Show that V is spanned by eigenvectors of T with eigenvalues r and s.
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
Let T: Rr - be a linear operator such that ToT Id Show that there is a basis B &Trelative to the basis B {ui , , , , , щ, vı , . . . ,VJofR" such that the representing matrix T Ul,. .. ,ur, Vi, has the form wherer +s-n(r or smay be zero), ie., adiagonal matrix whose diagonal entries are all Let T: Rr - be a linear operator such that ToT Id Show that there is...
1. let V be a vector space and T an operator on V (i.e., a linear map T: V--> V). Suppose that T^2 - 5T +6I = 0, where I is the identity operator and 0 stands for the zero operator ... Read Section 3.E and 3.F V) 1. Let V be a vector space and T an operator on V (i.e., a linear map T: V -» Suppose that T2 - 5T + 61 = 0, where I is...
What complex numbers might be eigenvalues of a linear operator T such that (a) T-I, (b) T2 - 5T6 0?
6) Show that if L is a linear operator, ie. L(g/t Gf,)-G L(J) +GJ (G) , then 1 (gf, + c2f2 + c3f3)-Q. (f) + GL(7) + c,L(左) Hint: you might find it easier to begin with the right hand side of this expression. You don't have to prove this but it should be obvious after you're done that this extends inductively to a linear combination of any number of functions
Suppose that a linear operator T on a complex vector space with an inner product, has minimal polynomial 2 + (1 + i)z + 7i. Find the minimal polynomial of the adjoint operator T*. Justify your answer.
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.