All this would be much shorter if you knew what the traspose of a Linear Transformation is, and its properties.
The proof would go as follows. Let be the standard basis for (i.e. ). This is an orthonormal basis, so for each (1)
Let us compute . By (1) we have: thus:
We will prove that holds if and only if
Since you can check it maps an orthonormal basis into an orthonormal basis. Thus is an orthonormal basis.
Now: and hence
Thus it follows that: and therefore: and the proof of the direct is complete
Step 1: Since then exists and
Step 2: , by the symmetry of the inner-products over the filed of the real numbers : and step 1 implies hence: for all (2)
Step 3. Prove that (2) implies for all
Step 4: for all and the proof is complete