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Q2.3-30Q

Expert-verifiedFound in: Page 93

Book edition
5th

Author(s)
David C. Lay, Steven R. Lay and Judi J. McDonald

Pages
483 pages

ISBN
978-03219822384

**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.

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

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

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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