A generalisation of Gram-Schmidt’s orthonogonalisation process

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.

Published by ayushkhaitan3437

Hello! My name is Ayush Khaitan, and I'm a graduate student in Mathematics. I am always excited about talking to people about their research. Please please set up a meeting with me if you feel that I might have an interesting perspective to offer- https://calendly.com/ayushkhaitan/meeting-with-ayush

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