An Application of Linear Algebra |

Kevin McGoldrick - Math 345, Fall 2009 |

Least Squares Estimation The Necessary Linear Algebra |

Here we present some results allowing us to derive a formula for χ. Theorem 1: Suppose V is a finite dimensional inner product space, with subspace W. Let u be any vector in V. The “best approximation” from u to W is the orthogonal projection of u onto W, denoted proj ||u-proj Proof: For any w in W, we have u-w = (u-(proj Note that ((proj (u-(proj ||u-w|| = ||(u-(proj
Since ||((proj ||u-w|| so ||u-w|| > ||(u-(proj as desired. |

Here, P is the orthogonal projection of V onto the plane. (P = proj |