Check whether a function is an isometry using definition

Question

Given the definition that a function $f: V\rightarrow W$ is an isometry iff the image of an orthonormal basis of V is an orthonormal basis of W, how do I know if matrix $$A=\begin{bmatrix}\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}\end{bmatrix}$$ is an isometry if I don't know V and W?

Answer 1

If $V$ and $W$ are not specified, you may assume that $V=W=\Bbb R^2.$

Note that the given transformation is a $45$ degree counterclockwise rotation, which preserves the angles and distances; hence, it is an isometry.