Find an example of a vector space V, and a linear transformation T : V +...
3. (6 marks) Find an example of a vector space V, and a linear transformation T : V + V such that R(T) = ker(T). Your vector space V must have dimension > 2. You may find it helpful to let V be a euclidean space and T a matrix transformation, but that is not necessary. You must explain why your example works.
3. (6 marks) Find an example of a vector space V, and a linear transformation T : V +V such that R(T) = ker(T). Your vector space V must have dimension > 2. You may find it helpful to let V be a euclidean space and T a matrix transformation, but that is not necessary. You must explain why your example works.
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 :...
I need the answer to problem 6 Clear and step by step please Problem 4. Let V be a vector space and let T : V → V and U : V → V be two linear transforinations 1. Show that. TU is also a linear transformation. 2. Show that aT is a linear transformation for any scalar a. 3. Suppose that T is invertible. Show that T-1 is also a linear transformation. Problem 5. Let T : R3 →...
W is a rele that A linear transformation T from a vector space V into a vector space assigns to each vector 2 in V a unique vector T() in W. such that (1) Tutu = Tu+Tv for all uv in V, and (2) Tſcu)=cT(u) for all u in V and all scalar c. *** The kernel of T = {UE V , T(U)=0} The range of T = {T(U) EW , ue V } Define T :P, - R...
3. This example hopes to illustrate why the vector spaces the linear transformation are defined on are critical to the question of invertibility. Let L : → p, be defined by L(p)(t+1)p(t)-plt). (a) Given a basis of your choice, find a matrix representation of I with respect to your chosen basis (b) Show L: P+P is not invertible (e) Let V-span+21-4,+2t-8). It can be shown that L VV. Given an ordered basis for V of your choice, find a matrix...
(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,...
4.10. Let T be a linear transformation on a vector space V satisfying T-T2 = id. Show that T is invertible.
P.3.31 Let V be a complex vector space. Let T : Mn → V be a linear transformation such that T(XY) = T(YX) for all X, Y E Mn. Show that T(A) = (trA)T(In) for all A EM, and dim ker T = n² – 1. Hint: A = (A - tr A)) + tr A)In.
(1 point) Let V be a vector space, and T:V → V a linear transformation such that T(2ū1 – 3ū2) = 4ū1 + 5ū2 and T(-301 + 502) = 501 - 502. Then 1 T(0) = ปี1+ 1 T(02) = + T(481 - 423) = U1+ 1+ ว.