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Q2.3-30Q

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Linear Algebra and its Applications
Found in: Page 93
Linear Algebra and its Applications

Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

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Short Answer

If A is an \(n \times n\) matrix and the transformation \({\bf{x}}| \to A{\bf{x}}\) is one-to-one, what else can you say about this transformation? Justify your answer.

The transformation is invertible.

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: State the invertible matrix theorem

The statements are identical according to the invertible matrix theorem, as shown below:

  1. The matrix is invertible.
  2. For some b, the matrix equation \[A{\bf{x}} = {\bf{b}}\] has no unique solution (more than one solution).
  3. The linear transformation \(x| \to Ax\) is one-to-one.
  4. The mapping of \({\mathbb{R}^n}\) onto \({\mathbb{R}^n}\) is equivalent to the linear transformation \(x| \to Ax\).

Step 2: Define the transformation

According to the invertible matrix theorem, if the matrix equation \[A{\bf{x}} = {\bf{b}}\] has more than one solution, the transformation is one-to-one.

From the given statement, the linear transformation \(x| \to Ax\) is one-to-one. So, the matrix equation must have more than one solution for some b, and matrix A should be invertible.

Therefore, both the matrix and the transformation are invertible.

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