How can I proof if V a finite dimensional vector space over a field, and T is a linear transformation V->V,
V is cyclic relative to T iff minimal polynomial of T is equal to the characteristic polynomial of T?
How can I proof if V a finite dimensional vector space over a field, and T...
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.
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) Prove that W, is the set of all vector8 α in V such that (T-cd)"a-0 for some positive integer 'n (which may depend upon α). (b) Prove that the dimension of W, is di. (Hint: If T, is the operator induced on Wi by T, then...
Let V be a finite-dimensional complex vector space and let T from V to V be a linear transformation. Show that V is the direct sum of U and W where W and U are T-invariant subspaces and the restriction of T on U is nilpotent and the restriction of T on W is an isomorphism.
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 :...
7.3.1 Let U be a finite-dimensional vector space over a field F and T є L(U). Assume that λ0 E F is an eigenvalue of T and consider the eigenspace Eo N(T-/) associated with o. Let. uk] be a basis of Evo and extend it to obtain a basis of U, say B = {"l, . . . , uk, ul, . . . ,叨. Show that, using the matrix representation of T with respect to the basis B, the...
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...
Question 1. Let V be a finite dimensional vector space over a field F and let W be a subspace of Prove that the quotient space V/W is finite dimensional and dimr(V/IV) = dimF(V) _ dimF(W). Hint l. Start with a basis A = {wi, . . . , w,n} for W and extend it to a basis B = {wi , . . . , wm, V1 , . . . , va) for V. Hint 2. Our goal...
Let V and W be finite dimensional vector spaces and let T:V → W be a linear transformation. We say a linear transformation S :W → V is a left inverse of T if ST = Iy, where Iy denotes the identity transformation on V. We say a linear transformation S:W → V is a right inverse of T if TS = Iw, where Iw denotes the identity transformation on W. Finally, we say a linear transformation S:W → V...
Problem #6. Let V be a finite dimensional vector space over a field F. Let W be a subspace of V. Define A(W) e Vw)Vw E W). Prove that A(W) is a subspace of (V).
de(V)e(A) 6· Lot V be a finite dimensional vector space, and T : V → V be a linear oper- ator. Suppose that T2-Iv, the identity operator. Prove that T is diagonalizable and that 1 and1 are the only possible eigenvalues of T