Prove that the transformation T over a unit vector is also a unit vector

Question

Let the $T : \mathbb{R}^2 \to \mathbb{R}^2$ be a linear transformation. How can I prove that the transformation T over a unit vector is also a unit vector? Does a linear transformation always preserve the distance?

I know that there must exist a matrix $A$ such that $T(u) = A u$ and $||u||^2 = u \cdot u =1$.

$$||T(u)||^2 = T(u) \cdot T(u) = Au \cdot Au$$

Not sure how to finish this.

Answer 1

Linear transformations are affine, which means they preserve colinearity, parallelism, and origin, but they need not preserve distance. For example the matrix given by \begin{bmatrix} k & 0\\ 0 & 1 \end{bmatrix} Corresponds to the map $T:\mathbb{R}^2\to\mathbb{R}^2$ which stretches everything by $k$ along the $x$ axis. You can verify that this does not preserve unit vectors by considering the unit vector along the $x$ axis (unless $k = \pm 1$).

Answer 2

It’s not true in general. Counter example: $Tx=2x$.