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Expert-verified Found in: Page 93 ### 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 # 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.

See the 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 $A{\bf{x}} = {\bf{b}}$ 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. ### Want to see more solutions like these? 