Automorphism of R3
Task number: 4527
Decide whether the mapping \(f:\mathbb{R}^3 \to \mathbb{R}^3\) given by the formula
\(f (x, y, z) = (x + y - 2z, y - z, x - y)^\mathsf{T}\)
is an isomorphism of \(\mathbb{R}^3\) onto itself (so-called automorphism).
Solution
The matrix of the mapping with respect to the standard basis \(E\) of the space \(\mathbb R^3\) (on both sides) is: \[f_{E,E}= \begin{pmatrix} 1 & 1 & -2\\ 0 & 1 & -1\\ 1 & -1 & 0 \end{pmatrix} \]
By elementary transformations: \[\begin{pmatrix} 1 & 1 & -2\\ 0 & 1 & -1\\ 1 & -1 & 0 \end{pmatrix} \sim \begin{pmatrix} 1 & 1 & -2\\ 0 & 1 & -1\\ 0 & -2 & 2 \end{pmatrix} \sim \begin{pmatrix} 1 & 1 & -2\\ 0 & 1 & -1\\ 0 & 0 & 0 \end{pmatrix} \] we find that its rank is 2 and is therefore singular.
Answer
The mapping \(f\) is not an automorphism.
