Orthogonal Sets:
A set of vectors in is called orthogonal set, if . if
Orthogonal Basis
An orthogonal basis for a subspace W of is a basis for W that is also an orthogonal set.
Let S = be the orthogonal basis for a W of is a basis for W that is also a orthogonal set. We need to calculate such that :
Let’s take the dot product of u_1 both side.
Since, this is orthogonal basis . This gives :
We can generalize the above equation
Orthogonal Projections
Suppose {u_1, u_2,… u_n} is an orthogonal basis for W in . For each y in W:
Let’s take is an orthogonal basis for and W = span . Let’s try to write a write y in the form belongs to W space, and z that is orthogonal to W.
where
and
[Tex]y= \hat{y} + z[/Tex]
Now, we can see that z is orthogonal to both and such that:
Orthogonal Decomposition Theorem:
Let W be the subspace of . Then each y in can be uniquely represented in the form:
where is in W and z in W^{\perp}. If is an orthogonal basis of W. then,
thus:
Then, is the orthogonal projection of y in W.
Best Approximation Theorem
Let W is the subspace of , y any vector in . Let v in W and different from . Then also in W.
is orthogonal to W, and also orthogonal to . Then y-v can be written as:
Thus:
Thus, this can be written as:
and
References: