Orthogonal sets - Orthogonality and Least Squares

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Orthogonal sets


A set of vectors {v1,,vnv_1,\cdots,v_n} in Rn\Bbb{R}^n are orthogonal sets if each pair of vectors from the set are orthogonal. In other words,
vivj=0v_i \cdot v_j =0
Where iji \neq j.

If the set of vectors {v1,,vnv_1,\cdots,v_n} in Rn\Bbb{R}^n is an orthogonal set, then the vectors are linearly independent. Thus, the vectors form a basis for a subspace SS. We call this the orthogonal basis.

To check if a set is an orthogonal basis in Rn\Bbb{R}^n, simply verify if it is an orthogonal set.
y=c1v1+c2v2++cpvpy=c_1 v_1+c_2 v_2+\cdots+c_p v_p

Are calculated by using the formula:
ci=yvivivic_i = \frac{y \cdot v_i}{v_i \cdot v_i}
where i=1,,pi=1,\cdots,p.

A set {v1,,vpv_1,\cdots,v_p}is an orthonormal set if it's an orthogonal set of unit vectors.

If SS is a subspace spanned by this set, then we say that {v1,,vpv_1,\cdots,v_p} is an orthonormal basis. This is because each of the vectors are already linear independent.

A m×nm \times n matrix UU has orthonormal columns if and only if UTU=IU^T U=I.

Let UU be an m×nm \times n matrix with orthonormal columns, and let xx and yy be in Rn\Bbb{R}^n. Then the 3 following things are true:
1) Ux=x\lVert Ux \rVert = \lVert x \rVert
2) (Ux)(Uy)=xy (Ux) \cdot (Uy)=x \cdot y
3) (Ux)(Uy)=0(Ux) \cdot (Uy)=0 if and only if xy=0x \cdot y =0

Consider LL to be the subspace spanned by the vector vv . Then the orthogonal projection of yy onto vv is calculated to be:
y^=\hat{y}=projLy=yvvvv_Ly=\frac{y \cdot v}{v \cdot v}v

The component of yy orthogonal to vv (denoted as zz) would be:
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Orthogonal sets

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