Linear Transformation Orthogonality

Question

True or False:

If $T$ is a linear transformation from $R^n$ to $R^n$

such that

$$T\left(\vec{e_1}\right), T\left(\vec{e_2}\right), \ldots, T\left(\vec{e_n}\right) $$

are all unit vectors

then $T$ must be an orthogonal transformation

The answer is ? I know a linear transformation is orthogonal if it preserves the length of vectors. And I understand a linear transformation is orthogonal if $$T\left(\vec{e_1}\right), T\left(\vec{e_2}\right), \ldots, T\left(\vec{e_n}\right) $$ form an orthonomal basis.

But how do I prove that withough knowing what the $T$, transformaton, does to the elementary column vectors...? Could anybody help me prove this?

Answer 1

False. Let $T$ be the linear transformation represented by the matrix whose first row is all 1's, and the other rows are all 0's. $T(e_i)=e_1$ for all $i$.

Answer 2

Let $T e_k = \frac{1}{\sqrt{k}} \sum_{i=1}^k e_i$. Then $T$ is invertible, and $\|T e_k \| = 1$ for all $k$.