# Orthogonal Projections

• Last Updated : 16 Oct, 2021

### 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:

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