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Math Help - Orthogonal Projections

  1. #1
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    Orthogonal Projections

    Find the 2x2 matrix A of the orthogonal projection onto the line L in R^2 that consists of all scalar multiples of the vector

    (5, 1)
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  2. #2
    Super Member PaulRS's Avatar
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    Remember that in general, given a linear space V and a subspace S with <br />
\dim \left( S \right) =n<  + \infty <br />
the orthogonal projection of a vector \bold{v} over S is given by <br />
P_S \left( \bold{v} \right) = \sum\limits_{k = 1}^n {\left\langle {\bold{v},\bold{w}_k } \right\rangle  \cdot \bold{w}_k } <br />
where <br />
\left\{ {\bold{w}_1 ,...,\bold{w}_n } \right\} <br />
is an orthonormal base of S ( Such a base exists by Gram-Schmidt)

    So in our case let <br />
\bold{w} = \tfrac{1}<br />
{{\sqrt {26} }} \cdot \left( {\begin{array}{*{20}c}<br />
   5 & 1  \\<br /> <br />
 \end{array} } \right)<br /> <br />
this vector belongs to the line L and its norm is 1, thus ( since the dimension of the line is 1) it forms an orthonormal base of L.

    Thus: <br />
P_L \left( \bold{u} \right) = \left\langle {\bold{u},\bold{w}} \right\rangle  \cdot \bold{w}<br />
from there: <br />
P_L \left( {\begin{array}{*{20}c}<br />
   x & y  \\<br /> <br />
 \end{array} } \right) = \tfrac{1}<br />
{{26}} \cdot \left( {\begin{array}{*{20}c}<br />
   {25x + 5y} & {5x + y}  \\<br /> <br />
 \end{array} } \right)<br />
and note that this can be re-written as (with column vectors): <br />
\left( {\begin{array}{*{20}c}<br />
   {\tfrac{{25}}<br />
{{26}}} & {\tfrac{5}<br />
{{26}}}  \\<br />
   {\tfrac{5}<br />
{{26}}} & {\tfrac{1}<br />
{{26}}}  \\<br /> <br />
 \end{array} } \right) \cdot \left( {\begin{array}{*{20}c}<br />
   x  \\<br />
   y  \\<br /> <br />
 \end{array} } \right) = P_L \left( {\begin{array}{*{20}c}<br />
   x  \\<br />
   y  \\<br /> <br />
 \end{array} } \right)<br />
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