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3. (6 marks) Find an example of a vector space V, and a linear transformation T...
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.
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,
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 →...
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 :...
3. [20 marks] A linear transformation T: P2 + R’ is defined by [ 2a – b 1 T(a + bt + ct?) = a +b – 3c LC-a ] (1). [6 marks] Determine the kernel Ker T of the transformation T and express it in the form of a span of basis. Further, state the dimension of Ker T (2). [6 marks) Find the range Range T of the transformation T and express the range in the form of...
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,...
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...
Let L in R 3 be the line through the origin spanned by the vector v = 1 1 3 . Find the linear equations that define L, i.e., find a system of linear equations whose solutions are the points in L. (7) Give an example of a linear transformation from T : R 2 → R 3 with the following two properties: (a) T is not one-to-one, and (b) range(T) = ...
4.10. Let T be a linear transformation on a vector space V satisfying T-T2 = id. Show that T is invertible.