Guided Proof Let and beone-to-one linear transformations.

Chapter 6, Problem 6.206

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Guided Proof Let and beone-to-one linear transformations. Prove that thecomposition is one-to-one and thatexists and is equal toGetting Started: To show that is one-to-one, use thedefinition of a one-to-one transformation and show thatimplies For the second statement,you first need to use Theorems 6.8 and 6.12 to show thatis invertible, and then show that andare identity transformations.(i) Let Recall thatfor all vectors Now use the fact that and areone-to-one to conclude that(ii) Use Theorems 6.8 and 6.12 to show thatand are all invertible transformations. So,and exist.(iii) Form the composition It is alinear transformation from into To show that itis the inverse of you need to determine whetherthe composition of with on both sides gives anidentity transformation.