Orthogonal sets

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Intros
Lessons
  1. Orthogonal Sets Overview:
  2. Orthogonal Sets and Basis
    • Each pair of vector is orthogonal
    • Linear independent → Form a Basis
    • Calculate weights with Formula
  3. Orthonormal Sets and Basis
    • Is an orthogonal set
    • Each vector is a unit vector
    • Linear independent → Form a Basis
  4. Matrix UU with Orthonormal columns and Properties
    UTU=IU^T U=I
    • 3 Properties of Matrix UU
  5. Orthogonal Projection and Component
    • Orthogonal Projection of yy onto vv
    • The component of yy orthogonal to vv
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Examples
Lessons
  1. Orthogonal Sets and Basis
    Is this an orthogonal set?
    Is this an orthogonal set
    1. 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.
      1. 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?
        1. 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
          1. 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.