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...
Prob le m 5 (Bonus 2 points) Let V be a finite dimensional vector space. Suppose that T : V -» V is matrix representation with respect to every basis of V. Prove that the dimension of linear transform ation that has the same that T must be a scalar multiple of the identity transformation. You can assume V is 3 Prob le m 5 (Bonus 2 points) Let V be a finite dimensional vector space. Suppose that T :...
Let f: V W be a linear map. Prove that f(0) = 0, i.e., any linear transformation maps zero vector to zero vector.
Let V be a vector space, let S, T L(V), and assume that ST = TS. Prove that if ˇ V is an eigenvector for T with eigenvalue λ, then λ is also an eigenvalue for S Find an eigenvector for λ with respect to S, and prove your answer is correct. Let V be a vector space, let S, T L(V), and assume that ST = TS. Prove that if ˇ V is an eigenvector for T with eigenvalue...
A projection is a nonzero linear operator P such that P2-P. Let v be an eigenvector with eigenvalue λ for a projection P, what are all possible values of X? Show that every projection P has at least one nonzero eigenvector. A projection is a nonzero linear operator P such that P2-P. Let v be an eigenvector with eigenvalue λ for a projection P, what are all possible values of X? Show that every projection P has at least one...
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...
(N-copies). Prove that the = Cx Cx 2. Let V denote the vector space V operator T: V V defined by T(a1, a2,...)= (0, a1, a2, . ..) has no (nonzero) eigenvectors
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: C4 C4 be the linear operator such that 18= [ 0 1 0 1] 1 1 1 0 1-100 i Oi i 1 where 8 is the standard basis for C4. Let W be the T-cyclic subspace of C4 generated by w = (1,0,0,0). 3.1 Find the T-cyclic basis for W generated by w. 3.2 Find a basis for Wt. 3.3 Show that wt is T*- invariant.
for a linear operator T ∈ L(V), V is finite-dimensional. let C={r(T)(v): r(x) ∈ F[x], v non zero} show that C is an invariant of T for the subspace of V.