The approach to project out certain vector

[onako]

Given that certain nullspace is spanned by a vector v, what would be the procedure to project out the v component from certain vector u? Perhaps with the Gram-Schmidt process of orthonormalization by updating
v = v - proj(u, v)

where proj(u, v) = (<u, v>/<u, u>)u, and <u, v> denotes the inner product? If that is correct way to do it, please let me know. However, if there are alternatives, I would be happy to consider those.

 

[chiro]

Hey onako.

Yes, you're right on the money with using Gram-Schmidt process which basically does exactly what you are trying to do by 'projecting out' all of the components that have been calculated that correspond to elements of your new orthonormal basis.

Basically it boils down to thinking about the projection really is and this boils down to standard vector geometry definitions.

We know from Gram-Schmidt that if we have a vector v with u as our chosen first component of our new basis, then we do what you have said in your formula above. It is the best way to do this and it is the standard way to attack these kinds of problems.

The only thing I wanted to add was that if you need an orthonormal basis, just be aware to normalize but this is not a strict requirement for having a basis.

 

[onako]

Thanks. Indeed, I'll need a normalization; good that you pointed that out.