User:Eml5526.s11.team5.JA/Mtg18

 Mtg 18: Fri, 11 Feb 11

[[media: Fe1.s11.mtg18.djvu| Page 18-1 ]]


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- Gram matrix HW 2.7 [[media: Fe1.s11.mtg10.djvu| page 10-3 ]]:   $$ \ det(\boldsymbol{\Gamma}) \underbrace{=}_{\color{Blue} page 18-3} \Theta(10^{-12}) {\color{Red} \ne 0 } \ $$

$$
 *  $$ \displaystyle {\color{Red}(1)}
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Compute $$ \boldsymbol{\Gamma}^{-1} \ $$ (Team 2, WA p. 18-4)

$$
 *  $$ \displaystyle {\color{Red}(2)}
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HW 2.9 p.12-1 $$ \phi = \frac{\pi}{2} \ $$ 'singularity' (Team 4 and Team 5)


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$$ F_1(\phi) = \left \{ cos(jx + \phi), j = 0,1,2... \right \} =  \left \{  cos(\phi), cos(x+\phi), cos(2x+\phi), cos(3x+\phi), ... \right \} \ $$

$$
 *  $$ \displaystyle {\color{Red}(3)}
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$$ F_1(0) =  \left \{  cos(0) = {\color{Red}1}, cos(x), cos(2x), cos(3x), ... \right \} \ $$

$$
 *  $$ \displaystyle {\color{Red}(4)}
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$$ F_1(\frac{\pi}{4}) = \left \{  cos(\frac{\pi}{4}), cos(x+\frac{\pi}{4}), cos(2x+\frac{\pi}{4}), cos(3x+\frac{\pi}{4}), ... \right \} \ $$

$$
 *  $$ \displaystyle {\color{Red}(5)}
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$$ \underbrace{F_1(\frac{\pi}{2})}_{\color{Red}not \ linearly \ independent} =  \left \{  \underbrace{cos(\frac{\pi}{2}) = {\color{Red} 0} }_{\color{Blue} b_1(x)}, \underbrace{cos(x+\frac{\pi}{2})}_{\color{Blue} \underbrace{-sin(x)}_{b_2 (x)}}, \underbrace{cos(2x+\frac{\pi}{2})}_{\color{Blue} \underbrace{-sin(2x)}_{b_3 (x)}},... \right \} \  $$

$$
 *  $$ \displaystyle
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$$ \boldsymbol{\Gamma}(F_1(\frac{\pi}{2}) = [< \underbrace {b_{\color{Blue} i}}_{\color{Blue} i = row }, \underbrace{b_{\color{Red} j}}_{\color{Red} j = column }>] =

\begin{bmatrix} {\color{Red}0}     & {\color{Red}0}           & \cdots & {\color{Red}0}       \\ {\color{Red}0}     & \Gamma_{22} & \cdots  &  \\ \vdots & \vdots & & \\ {\color{Red}0}    &   &  & \end{bmatrix}

$$

$$
 *  $$ \displaystyle
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$$ \Rightarrow det \boldsymbol{\Gamma} = 0 \ $$

$$
 * <p style="text-align:right"> $$ \displaystyle
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[[media: Fe1.s11.mtg18.djvu| Page 18-2 ]]


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[[media: Fe1.s11.mtg10.djvu| (1)-(2) page 10-3 ]] & [[media: Fe1.s11.mtg16.djvu| (2)  page 16-2 ]]

<span id="(1)">
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$$ \tilde{k}(w^h,u^h) = \sum_i \sum_j c_i \underbrace{\tilde{k}(b_i,b_j)}_{\color{Blue} \tilde{K}_{ij} } d_j \ $$

$$
 * <p style="text-align:right"> $$ \displaystyle {\color{Red}(2)}
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, where

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$$ {\color{Blue} \tilde{K}_{ij} } = \tilde{k}(b_i,b_j) = \int_{\alpha}^{\beta} b_i{\color{Red} '} a_2 b_j{\color{Red} '} dx \ $$

$$
 * <p style="text-align:right"> $$ \displaystyle {\color{Red}(3)}
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[[media: Fe1.s11.mtg18.djvu| Page 18-3 ]]

From Team 4, HW 2.7 Input code: det( { {1,1/2,1/3,1/4,1/5}, {1/2, 1/3, 1/4, 1/5, 1/6}, {1/3, 1/4, 1/5, 1/6, 1/7}, {1/4, 1/5, 1/6, 1/7, 1/8}, {1/5, 1/6, 1/7, 1/8, 1/9} } )

Direct link with answer: http://www.wolframalpha.com/input/?i=det(+%7b+%7b1%2c1%2f2%2c1%2f3%2c1%2f4%2c1%2f5%7d%2c+%7b1%2f2%2c+1%2f3%2c+1%2f4%2c+1%2f5%2c+1%2f6%7d%2c+%7b1%2f3%2c+1%2f4%2c+1%2f5%2c+1%2f6%2c+1%2f7%7d%2c+%7b1%2f4%2c+1%2f5%2c+1%2f6%2c+1%2f7%2c+1%2f8%7d%2c+%7b1%2f5%2c+1%2f6%2c+1%2f7%2c+1%2f8%2c+1%2f9%7d+%7d+)+&incParTime=true



[[media: Fe1.s11.mtg18.djvu| Page 18-4 ]]

Input code: inv( { {1,1/2,1/3,1/4,1/5}, {1/2, 1/3, 1/4, 1/5, 1/6}, {1/3, 1/4, 1/5, 1/6, 1/7}, {1/4, 1/5, 1/6, 1/7, 1/8}, {1/5, 1/6, 1/7, 1/8, 1/9} } )

Direct link with answer: http://www.wolframalpha.com/input/?i=inv(+%7b+%7b1%2c1%2f2%2c1%2f3%2c1%2f4%2c1%2f5%7d%2c+%7b1%2f2%2c+1%2f3%2c+1%2f4%2c+1%2f5%2c+1%2f6%7d%2c+%7b1%2f3%2c+1%2f4%2c+1%2f5%2c+1%2f6%2c+1%2f7%7d%2c+%7b1%2f4%2c+1%2f5%2c+1%2f6%2c+1%2f7%2c+1%2f8%7d%2c+%7b1%2f5%2c+1%2f6%2c+1%2f7%2c+1%2f8%2c+1%2f9%7d+%7d+)+&incParTime=true



Direct Link to HW: http://en.wikiversity.org/w/index.php?title=User:Eml5526.s11.team4/HW2#Problem_2.7_Verifying_the_orthogonality_of_the_family