Orthogonal projections

Lessons

• Introduction
  Orthogonal Projections Overview:
  a)

  The Orthogonal Decomposition Theorem
  • Make $y$ as the sum of two vectors $\hat{y}$ and $z$
  • Orthogonal basis → $\hat{y}= \frac{y \cdot v_1}{v_1 \cdot v_1}v_1 + \cdots + \frac{y \cdot v_p}{v_p \cdot v_p}v_p$
  • Orthonormal basis → $\hat{y}=(y\cdot v_1)v_1+\cdots +(y\cdots v_p)v_p$
  • $z=y - \hat{y}$
  
  
  b)

  Property of Orthogonal Projections
  • proj$_s y=y$
  • Only works if $y$ is in $S$
  
  
  c)

  The Best Approximation Theorem
  • What is the point closest to $y$ in $S$? $\hat{y}$!
  • Reason why: $\lVert y - \hat{y} \rVert$ < $\lVert y-u \rVert$
  • The Distance between the $y$ and $\hat{y}$
  
  

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• 1.
  The Orthogonal Decomposition Theorem
  Assume that {$v_1,v_2,v_3$} is an orthogonal basis for $\Bbb{R}^n$. Write $y$ as the sum of two vectors, one in Span{$v_1$}, and one in Span{$v_2,v_3$}. You are given that:
  

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• 2.
  Verify that {$v_1,v_2$} is an orthonormal set, and then find the orthogonal projection of $y$ onto Span{$v_1,v_2$}.
  

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• 3.
  Best Approximation
  Find the best approximation of $y$ by vectors of the form $c_1 v_1+c_2 v_2$, where , and , .

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• 4.
  Finding the Closest Point and Distance
  Find the closest point to $y$ in the subspace $S$ spanned by $v_1$ and $v_2$.

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• 5.
  Find the closest distance from $y$ to $S=$Span{$v_1,v_2$} if
  

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