Gram-Schmidt process

The Gram-Schmidt process is a way of converting one set of vectors that forms a basis into another, more friendly one.

Suppose we have a set of vectors $\{v_{1},...,v_{n}\}$ that form a basis for $R^{n}$, and that we wish to convert these into a "friendly" basis which is easier to work with later. We begin by finding out which component of a vector is being unfriendly to a second vector, which we may do with inner products. If we have two vectors $u,v$, then we may find the component of $u$ being unfriendly to $v$ with $proj_{u}(v)={\frac {<v,u>}{<u,u>}}u$. By subtracting this from $v$, we get the component friendly to $u$. By returning to our abstract set $\{v_{1},...,v_{n}\}$, we may make use of this observation to construct a general algorithm to convert an arbitrary basis into a friendly basis. Like from our original set, each time we use the process on a new vector, it is guaranteed to be mutually friendly with all the previous vectors.

Define:

$w_{1}$ = $v_{1}$

$w_{2}$ = $v_{2}-proj_{w_{1}}v_{2}$

et cetera, with the general term:

$w_{n}=v_{n}-\sum _{i=1}^{n-1}{proj_{w_{i}}v_{n}}$

Should we wish to make this a friendly and cute basis, we can simply make each new element of the basis cute, by replacing both $v_{n}$ with ${\frac {v_{n}}{||v_{n}||}}$.