Originally Posted by

**Tikoloshe** I do not believe the OP is looking for orthonormality of the images of the original basis vectors, only orthogonality, so the result can be (in fact is) true.

This comes down to finding the singular value decomposition. Basically, if $\displaystyle s_1,\ldots,s_n$ are the singular values of $\displaystyle T$, then there exist orthonormal bases $\displaystyle \{v_1,\ldots,v_n\}$ and $\displaystyle \{u_1,\ldots,u_n\}$ such that for an arbitrary $\displaystyle x\in V$, we can write $\displaystyle Tx=\sum_{i=1}^n s_i\langle x,v_i\rangle u_i$. One can obtain the singular value decomposition by taking the polar decomposition of $\displaystyle T$ and then applying the finite-dimensional spectral theorem to $\displaystyle \sqrt{T^\ast T}$.

Yes, I have a feeling the purpose of part 2 was to show the existence of a polar decomposition.