I have a vector v orthogonal to [w1 w2 w3]. Is it possible to get the projection of v to the space spanned by W=[w1 w2 w3 w4] without calculating the inverse of $\displaystyle W^tW$?
If v is already orthogonal to w1, w2, and w3, its projection onto that subspace is just its projection onto w4. I don't know if that involves "calculating the inverse of $\displaystyle W^tW$" because I don't know what "$\displaystyle W^tW$" means for W a vector space.
W is the basis of the space. I was thinking to get the projection of v to W, I need to use $\displaystyle W(W^tW)^{-1}W^tv$. Could you please give a brief proof of "If v is already orthogonal to w1, w2, and w3, its projection onto that subspace is just its projection onto w4". Sorry for my ignorance. I am not a math guy.