Let V be a vector space, let S, T L(V), and assume that ST = TS. Prove that if ˇ V is an eigenvec...
7.3 (Eigenvalues II) Let V be a vector space over K and let f,g E End(V). Show that: a) If-1 is an eigenvalue of ff, then 1 is an eigenvalue of f3. b) If u is an eigenvector off o g to the eigenvalue λ such that g(v) 0, then g(v) is an eigenvector of g o f. If, in addition, dim V < oo,then f o g and go f have the same eigenvalues c) If {ul, unt is...
1. Let V be a vector space with bases B and C. Suppose that T:V V is a linear map with matrix representations Ms(T)A and Me(T) B. Prove the following (a) T is one-to-one iff A is one-to-one. (b) λ is an eigenvalue of T iff λ is an eigenvalue of B. Consequently, A and B have the same eigenvalues (c) There exists an invertible matrix V such that A-V-BV 1. Let V be a vector space with bases B...
Let v 2 Rn be a unit vector. Define G = I ? vvT . (a) Show G is symmetric and G2 = G. (b) Prove v is an eigenvector, find the associated eigenvalue. (c) Prove that if < u; v >= 0 then u is also an eigenvector of G. (d) Prove that G is diagonalizable. Let v ER" be a unit vector. Define G=I - vt. (a) Show G is symmetric and G =G. (b) Prove v is...
(6) In each case V is a vector space, T: V- V is a linear transformation, and v is a vector in V. Determine whether the vector v is an eigenvector of T If so, give the associated eigenvalue Is v an eigenvector? If so, what is the eigenvalue? (b) T : M2(R) → M2(R) is given by [a+2b 2a +b c+d2d and V= Is v an eigenvector? If so, what is the eigenvalue? (c) T : R2 → R2,...
3. Let Te L(V), where V is a finite-dimensional C-vector space. Prove that T is diago- nalizable if and only if Ker(T – a id) n Im(T - a id) = {0} for all a E C.
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.
3. Let TEL(V,W), and assume that S E L(W) is an isometry. Prove that T and ST have the same singular values.
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...
10 9. Let U be a finite-dimensional vector space and TE LU). Prove the following statements. (a) (5 pts) Let λ be an eigenvalue T whose geometric multiplicity is m, and algebraic multiplicity is ma. Then (b) (5 pts) Let u be a cyclic vector of T of period k 2 2 (such that T*(u) 0 but T-(u) 0). Then are linearly independent. 10 9. Let U be a finite-dimensional vector space and TE LU). Prove the following statements. (a)...
(bonus) Prove that operator A: L²(0,1) - L’(0,1), Ar(t) = 5 ts(1 – st)x(s)ds is compact and find its spectrum.