Orthogonal sets

Orthogonal sets

Lessons

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:
z=yy^z=y-\hat{y}
  • 1.
    Orthogonal Sets Overview:
    a)
    Orthogonal Sets and Basis
    • Each pair of vector is orthogonal
    • Linear independent → Form a Basis
    • Calculate weights with Formula

    b)
    Orthonormal Sets and Basis
    • Is an orthogonal set
    • Each vector is a unit vector
    • Linear independent → Form a Basis

    c)
    Matrix UU with Orthonormal columns and Properties
    UTU=IU^T U=I
    • 3 Properties of Matrix UU

    d)
    Orthogonal Projection and Component
    • Orthogonal Projection of yy onto vv
    • The component of yy orthogonal to vv


  • 2.
    Orthogonal Sets and Basis
    Is this an orthogonal set?
    Is this an orthogonal set

  • 3.
    Verify that Verify that this is an orthogonal basis for R^2 is an orthogonal basis for R2\Bbb{R}^2, and then express express it as a linear combination of the set of vectors in B as a linear combination of the set of vectors in BB.

  • 4.
    Orthonormal Sets/Basis
    Is set BB is an orthonormal basis for R3\Bbb{R}^3?
    Is set B is an orthonormal basis for R^3?

  • 5.
    Let Matrix U, x, and y where UU has orthonormal columns and UTU=IU^TU=I. Verify that
    (Ux)(Uy)=xy(Ux)\cdot (Uy)=x\cdot y

  • 6.
    Orthogonal Projection
    Let vector y and vector v. Write yy as the sum of two orthogonal vectors, one in Span{vv} and one orthogonal to vv.