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Linear Algebra and its Applications
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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 equation \(A{\bf{x}} = {\bf{b}}\) has more than one solution for some b, then the transformation \({\bf{x}}| \to A{\bf{x}}\) is not one-to-one. What else can you say about this transformation? Justify your answer.a

The transformation is not invertible.

See the step by step solution

Step by Step Solution

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 \(A{\bf{x}} = {\bf{b}}\) does not have a 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 matrix equation has more than one solution for some b, but the linear transformation \(x| \to Ax\) is not one-to-one.

So, the given statement cannot be true.

Therefore, both the matrix and the transformation are not invertible.

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