Thread: Gram-Schmidt process vectors

Prove that if is an orthogonal set of nonzero vectors, then the vectors derived from the Gram-Schmidt process satisfy

my proof so far:

I intend to use induction.

Now, is travial.

Suppose that n = k is true, then I have

Now, how would I use that information as well as the fact that the vecters are orthogonal to get k+1 is true?

thanks

Originally Posted by tttcomrader

Prove that if is an orthogonal set of nonzero vectors, then the vectors derived from the Gram-Schmidt process satisfy

my proof so far:

I intend to use induction.

Now, is travial.

Suppose that n = k is true, then I have

Now, how would I use that information as well as the fact that the vecters are orthogonal to get k+1 is true?

This is easy if you use strong induction (in other words, assume that the inductive hypothesis holds for all n≤k, not just for n=k). The formula for v_{k+1} is . But if for 1≤j≤k then each term in that sum will vanish, because the w's are orthogonal to each other.