Guided Proof Let and be one-to-one linear transformations. Prove that the composition is

Chapter 6, Problem 77

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Guided Proof Let and be one-to-one linear transformations. Prove that the composition is one-to-one and that exists and is equal to Getting Started: To show that is one-to-one, you can use the definition of a one-to-one transformation and show that implies For the second statement, you first need to use Theorems 6.8 and 6.12 to show that is invertible, and then show that and are identity transformations. (i) Let Recall that for all vectors Now use the fact that and are one-to-one to conclude that (ii) Use Theorems 6.8 and 6.12 to show that and are all invertible transformations. So and exist. (iii) Form the composition It is a linear transformation from to To show that it is the inverse of you need to determine whether the composition of with on both sides gives an identity transformation.