Linear Algebra and Its Applications (5th Edition)

Published by Pearson
ISBN 10: 032198238X
ISBN 13: 978-0-32198-238-4

Chapter 2 - Matrix Algebra - 2.3 Exercises - Page 118: 30

Answer

Applying theorems 8 and 9, we find that $x\mapsto Ax$ is invertible

Work Step by Step

Apply Th.8. We are given that $A$ is a square $n\times n$ matrix, and $\mathrm{f}.\quad$ The linear transformation $x\mapsto Ax$ is one-to-one. is true. Then, $\mathrm{a}. \quad A$ is an invertible matrix. $\mathrm{i}.\quad$ The linear transformation $x\mapsto Ax$ maps $\mathbb{R}^{n}$ onto $\mathbb{R}^{n}$. are also true. So, A is invertible and $x\mapsto Ax$ maps $\mathbb{R}^{n}$ onto $\mathbb{R}^{n}$. By Th.9, the linear transformation $x\mapsto Ax$ is invertible.
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