(6) In each case V is a vector space, T: V- V is a linear transformation,...
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.
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,
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...
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 →...
7. In each case, evaluate T(v), using the given information, and assuming that T is a linear transformation (a) T(1.1.1) = (1.0.0), T(1,-1,-1) = (0,1,0), T(-1.-1.1) = (0,0.1), v = (3.2. 1). (b) T(a) e + sin r, T(b) = ?? - csr, (c) - 2+ - 31", v = 7a - 2b + c (c) Ta + b + c) - u T(2a) us. T(3b) , v a + 2b + 3c. (a) Téu + v) - (} ^)....
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...
4.10. Let T be a linear transformation on a vector space V satisfying T-T2 = id. Show that T is invertible.
Determine whether or not the following transformation T :V + W is a linear transformation. If T is not a linear transformation, provide a counter example. If it is, then: (i) find the nullspace N(T) and nullity of T, (ii) find the range R(T) and rank of T, (iii) determine if T is one-to-one, (iv) determine if T is onto. : (a) T: R3 + R2 defined by T(x, y, z) = (2x, y, z) (b) T: R2 + R2...
Let t be the linear transformation t: r2 -> r2 that reflects a vector about the line y=x. Find the eigenvalue and eigenvectors of T. How can you interpret this geometrically?