Gram Schmidt-The arts behind it

Question

One simple question,why Gram-Schmidt works? Honestly, I can't crack the key for this question given by my lecturer,so if you guys don't mind to share some thought, it will be very helpful.

Saludos

Answer 1

It is necessary (and maybe sufficient) that you have clear the conceprt behind first step.

Let's consider two linearly independent vector and then subtract from the second its vector projection on the first one. In this way you obtain 2 orthogonal vectors.

Then consider a third linearly independent vector and subtract from it its vector projection on the two orthogonal vectors you have. In this way you have obtained 3 orthogonal vectors and so on.

Answer 2

Suppose you have a given vector $v$ and a given direction. Let $u$ be any vector is the given direction. Is it possible to decompose $v$ into two orthogonal vectors, one of which is in the direction of $u?$ The answer is yes. Simply find the projection of $v$ over $u$ and and subtract it from $v$. Now Your $v$ looks like the hypotenuse of a right triangle with the base in the direction of $u$. The Gram-Schmidt process generalizes this concept step to higher dimensions.