Question

If Ais 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.

Short answer

The transformation is invertible.

Step-by-step solution

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 \(x| \to Ax\).

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.