# Orthogonal Projections

### 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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