Why are both the algorithm used for finding the orthogonal and orthonormal basis the same? I'm relying on a set of slides given by by lecturer (known to be sloppy!) and I want to confirm if it should be the same.
Edit:
As can be seen, the algorithm are both the same!
On
They are not entirely the same. The process for obtaining the orthogonal basis results in an orthogonal basis that is not necesseraly orthonormal.
In fact, one process is a subprocess in the other. If $B$ is a basis, then the Gram-Schmidt algorithm converts $B$ into $B'$ which is also a basis of the same space as $B$, but it is an orthogonal basis. Then, if you want an orthonormal base, you have to normalize each vector in $B'$ to obtain $B''$.
Edit:
Judging by your notes, it seems to me that your professor names both processes "the Gram-Schmidt process", but the algorithms are most certainly not the same, as the other algorithm ends with:
$$v_n' = \frac{v_n}{\sqrt{\langle v_n, v_n\rangle}}$$
An orthonormal basis is an orthogonal basis whose vectors have been scaled to have length $1$ under the norm induced by the inner product. That is, if you wish to find an orthogonal basis, it is enough to find an orthonormal one. Conversely, if you have an orthogonal basis, you may make it orthonormal by simply scaling each vector.