1. Because w is left arbitrary. It's true for ALL vectors w in the subspace W.
2. Because g is in W, and w is in W. Since W is a subspace, linear combinations of vectors in the subspace remain in the subspace.
Least Squares Approximation Theorem:
Let be continuous on , and let be a finite-dimensional subspace of . The least square approximating function of with respect to is given by
Proof:
To show that is the least squares approximating function of , prove that the inequality
is true for any vector in . By writing as
you can see that is orthogonal to each , which in turn implies that it is orthogonal to each vector in . In particular, is orthogonal to
-------------Proof Ends------------
My question has two parts.
First question:
Why proving
is enough for this proof? There might be other vectors that make the least square approximation smaller. Why choose particular orthonormal basis?
Second question:
What is the reason behind the implicit statement is in the subspace ?
What reasoning make ? Is there any theorem I can't remember now for this question?
1. Because w is left arbitrary. It's true for ALL vectors w in the subspace W.
2. Because g is in W, and w is in W. Since W is a subspace, linear combinations of vectors in the subspace remain in the subspace.
Thank you for your answers. I get the answer no. 2. But having difficulty in answer no. 1.
I'll tell you my understanding.
is not a generalized vector. is a orthonormal basis that are normalized and orthogonal to each other. Most vectors in subspace are not normalized and orthogonal. So why a particular class of vector speaks for all generalized vectors in subspace ? I don't understand that.
Can you kindly tell me what you meant by "left arbitrary"?
Hmm. Well, I could be wrong, but it looks to me like you might be confusing the arbitrary vector chosen at the beginning of the proof, and the basis vectors of the subspace They are not the same thing. It might have been clearer in the writing of the proof to pick an arbitrary vector instead of Does that clear things up a bit?