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"Template:What 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 $p r o j_{u} (v) = \frac{< v, u >}{< u, u >} u$ . 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:

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

$w_{2}$ = $v_{2} - p r o j_{w_{1}} v_{2}$

et cetera, with the general term:

$w_{n} = v_{n} - \sum_{i = 1}^{n - 1} p r o j_{w_{i}} v_{n}$

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

Gram-Schmidt process

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