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...
I need the answer to problem 4 (exercises 1, 2, 3) 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...
Please give answer with the details. Thanks a lot! Let T: V-W be a linear transformation between vector spaces V and W (1) Prove that if T is injective (one-to-one) and {vi,.. ., vm) is a linearly independent subset of V the n {T(6),…,T(ền)} is a linearly independent subset of W (2) Prove that if the image of any linearly independent subset of V is linearly independent then Tis injective. (3) Suppose that {b1,... bkbk+1,. . . ,b,) is a...
Let T: V + W be a linear transformation. Assume that T is one-to-one. Prove that if {V1, V2, V3} C V is a linearly independent subset of V, then {T(01), T(v2), T(13)} C W is a linearly independent subset of W.
Problem 1: Let W = {p(t) € Pz : p'le) = 0}. We know from Problem 1, Section 4.3 and Problem 1, Section 4.6 that W is a subspace of P3. Let T:W+Pbe given by T(p(t)) = p' (t). It is easy to check that T is a linear transformation. (a) Find a basis for and the dimension of Range T. (b) Find Ker T, a basis for Ker T and dim KerT. (c) Is T one-to-one? Explain. (d) Is...
7. Let T : V → W be a linear transformation, and let v1,v2,...,vn be vectors in V. Suppose that T (v1), T (v2), . . . , T (vn) are linearly independent. Show that v1, v2, . . . , vn are linearly independent.
Let V and W be vector spaces over F, and let f: V W be a linear transformation. (a) Prove that f is one-to-one if and only if f carries linearly independent (b) Suppose that f is one-to-one and that S is a subset of V. Prove that subsets of V to linearlv independent subsets of W S is linearly independent if and only if (S) is linearly independent.
Problem 1. The figure below shows the vectors u, v, and w, along with the images T(u) and T(v) to the right. Copy this figure, and draw onto it the image T(w) as accurately as possible. (Hint: First try writing w as a linear combination of u and v.) TV (u) Problem 2. Let u = | and v Suppose T : R2 + R2 is a linear transformation with 6 1 3) Tu = T(u) = -3 and T(v)...
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 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 9. Let V be a vector space over a field F (a) The empty set is a subset of V. Is a subspace of V? Is linearly dependent or independent? Prove your claims. (b) Prove that the set Z O is a subspace of V. Find a basis for Z and the dimension of Z (c) Prove that there is a unique linear map T: Z → Z. Find the matrix representing this linear map and the determinant of...