Gram-Schimidt process(trying to understand the projection operator)

Question

For Gram-Schmidt process, the projection operator is defined by $proj_u(v)=\frac{<v,u>}{<u,u>}u$. Can someone give me an reference for interpreting this projection operator? What do $<v,u>$ and $<u,u>$ represent geometrically? (Given that I have some basic knowledge of how inner product works). By the way, how do people come up with the Gram-Schmidt formula?

Answer 1

This figure might help clarify some of the details:

Note that the vector $v-proj_u\left(v\right)$ is orthogonal to $u$ in the figure, which is what you are trying to accomplish using Gram-Schmidt. This process essentially repeats as you include more vectors.

I hope this helps.