4. Let V = R4, and let W = Spa ermine whether the set is linearly...
1. Determine whether the following set is linearly independent or not. Prove your clas a. [1+1, 2+2-2,1 +32"} b. {2+1, 3x +3',-6 +2"} 8. Let T be a linear transformation from a vector space V to W over R. . Let .. . be linearly independent vectors of V. Prove that if T is one to one, prove that (un)....(...) are linearly independent. (m) is ) be a spanning set of V. Prove that it is onto, then Tu... h...
Let v1,v2,v3 and v4 be linearly independent vectors in R4. Determine whether each set of vectors is linearly independent or dependent. Please solve d) and f) U1, 2, 03, 4
suppose that s=(v1,v2,......vm) is a finite set of linearly independent vectors in V, and w ∈ V some other vector. Let T= S ∪ (W). Prove that T is not linearly independent if and only if w∈ span(s).
1· Let S {u,v) be a linearly, independent set. Prove that(u+ v.u-v) is linearly independent. 2. Let H :2y1. Prove that H is not a subspace of f2.
Let u = and v= Determine whether the vectors u and v are linearly independent or linearly dependent, and choose the most correct answer below. A. The vectors are linearly independent. B. We cannot easily tell whether the vectors are linearly independent or linearly dependent. C. The vectors are linearly dependent.
Let {v1, v2,v3} be a linearly independent set in R^n and let v = -αv3 +v1,w = v2 - αv1, u= v3-αv2 where αER, find all the values of α, where v, w, u are linearly dependent. do not use matrices.
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 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.
Let T :V → W be an isomorphism. Prove that if {ū1, ū2, ..., ūn} is a linearly independent set in W, then the preimages of {ū1, ū2, ... , ūn} is a linearly independent set in V.
Proble m 3. Let T: V ->W be (1) Prove that if T is then T(),... ,T(Fm)} is a linearly indepen dent subset of W (2) Prove that if the image of any linearly in depen dent subset of V is linearly indepen dent then T is injective (3) Suppose that {,... ,b,b^1,...,5} is Prove that T(b1), .. . , T(b,)} is a basis of im(T) (4) Let v1,. Vk} be T(v1),..,T(vk) span W lin ear transform ation between vector...