A generalisation of Gram-Schmidt’s orthonogonalisation process

by ayushkhaitan3437

I just read up about the Gram-Schmidt orthogonalization process.

Say we have \{b_1,b_2,\dots,b_r\} as an orthonormal basis for a subspace. Now let w=a-(a,b_1)b_1-(a,b_2)b_r-\ldots-(a,b_r)b_r for ANY a\in X, where X is the vector space. The set \{b_1,b_2,\dots,b_r,\frac{w}{\|w\|}\} is orthonormal.

$a$ needn’t be a vector from X\setminus S, where S is spanned by \{b_1,b_2,\dots,b_r\}.

We require a to be from X\setminus S only because we want to make the orthonormal set span a larger vector space.