TEST BANK FOR Manifolds, Tensor and Forms An Intro

Exam (elaborations) TEST BANK FOR Manifolds, Tensor and Forms An Intro 6 Let v1, v2 ∈ ker T . Then T (av1 + bv2) = aT v1 + bT v2 = 0, so ker T is closed under linear combinations. Moreover ker T contains the zero vector of V. All the other vector space properties are easily seen to follow, so ker T is a subspace of V. Similarly, let w1, w2 ∈ im T and consider aw1 + bw2. There exist v1, v2 ∈ V such that T v1 = w1 and T v2 = w2, so T (av1 + bv2) = aT v1 + bT v2 = aw1 + bw2, which show...