(N-copies). Prove that the = Cx Cx 2. Let V denote the vector space V operator...
Problem 4. Let n E N, and let V be an n-dimensional vector space. Let(, ,): V × V → R be an nner product on V (a) Prove that there exists an isomorphism T: V -R" such that (b) Is the isomorphism T you found in part (a) unique? Give a proof or a counterexample. (c) Let A be an n × n symmetric matrix such that T A > 0 for all nonzero ERT. Show that there exists...
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...
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...
Let V be the vector space of all sequences over R. Given (a1, a2, T,U V V by ) e V, define : ) ...) = (0, a1, 0, a2, 0, a3, . . . ) Тај, а2, аз, ад, 0, аз, (a1, a3, a5,.) and U(a1, a2, a3, a4, (a) Find N(T) and N(U) (b) Explain why T is onto, but not 1-1 (c) Explain why U is 1-1, but not onto.
QUESTION 5 Let V denote an arbitrary finite-dimensional vector space with dimension n E N Let B = {bi, bn} and B' = { bị, b, } denote two bases for V and let PB-B, be the transition matrix from B to B' Prove that where 1 V → V is the identity transformation, i e 1(v) v for all v E V Note that I s a linear transformation 14] QUESTION 5 Let V denote an arbitrary finite-dimensional vector...
3.2.1: Let V be a vector space with Basis B and let L be an operator on V. L2 means the operator applied twiceL2(v) = L(L(v)). Show that the Matrix of L2 is the square of the matrix of L, i.e LL? Demonstrate this problem for the space V span (1, t, t2, t3) and let L d/dt (so L2d/dt) SO
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.
row reduction in uncountable dimension. Part 2. (Row-reduction in countably-infinite dimension) Let V denote the vector space of polynomials (of all degrees). Recall that V is an infinite-dimensional vector space, but it has a countable basis. Consider Te Hom(V, V) defined as T(p())5p () 10p(x - 1) 2.1. Write T as an oo x oo matrix, in the standard basis 1,X, x2, 13,... of V 2.2. Write T as an oo x oo matrix, in the basis 1, + 1,...
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...
Problem 4. Let n E N. We consider the vector space R” (a) Prove that for all X, Y CR”, if X IY then Span(X) 1 Span(Y). (b) Let X and Y be linearly independent subsets of R”. Prove that if X IY, then X UY is linearly independent. (C) Prove that every maximally pairwise orthogonal set of vectors in R” has n + 1 elements. Definition: Let V be a vector space and let U and W be subspaces...