diff options
| -rw-r--r-- | šola/la/dn8/dokument.lyx | 107 |
1 files changed, 61 insertions, 46 deletions
diff --git a/šola/la/dn8/dokument.lyx b/šola/la/dn8/dokument.lyx index c01e7bc..6d8a154 100644 --- a/šola/la/dn8/dokument.lyx +++ b/šola/la/dn8/dokument.lyx @@ -725,7 +725,9 @@ Po predpostavki velja \begin_inset Formula $\left(AA^{*}-A^{*}A\right)^{*}=AA^{*}-A^{*}A$ \end_inset - + +\series bold +TODO TODO TODO XXX XXX XXX XXX XXX XXX TODO TODO TODO \end_layout \end_deeper @@ -804,25 +806,76 @@ Baza za \end_inset je -\begin_inset Formula $B=\left\{ v_{1},v_{2}\right\} =\left\{ \left(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right),\left(\frac{1}{2},\frac{-1}{2},\frac{-1}{2},\frac{-1}{2}\right)\right\} $ +\begin_inset Formula $B=\left\{ v_{1},v_{2}\right\} =\left\{ \left(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right),\left(\frac{1}{2},\frac{1}{2},\frac{-1}{2},\frac{-1}{2}\right)\right\} $ \end_inset . \end_layout \begin_layout Enumerate -Uporabimo fourierov razvoj -\begin_inset Formula $y$ +Dopolnimo +\begin_inset Formula $B$ \end_inset - po -\begin_inset Formula $B$ + do baze +\begin_inset Formula $\mathbb{R}^{4}$ \end_inset . + Dopolnitev +\begin_inset Formula $\left\{ u_{1},u_{2}\right\} $ +\end_inset + + bo ortonormirane baze za +\begin_inset Formula $W^{\perp}$ +\end_inset + +, nato uporabimo Fourierov razvoj po dopolnjeni bazi. + Bazo podprostora dopolnimo tako, da rešimo sistem enačb. +\begin_inset Formula +\[ +\left\langle \left(x_{1},y_{1},z_{1},w_{1}\right),\left(3,3,-1,-1\right)\right\rangle =0\quad\quad\quad\left\langle \left(x_{2},y_{2},z_{2},w_{2}\right),\left(1,1,1,1\right)\right\rangle =0 +\] + +\end_inset + + +\begin_inset Formula +\[ +\left[\begin{array}{cccc} +1 & 1 & 1 & 1\\ +3 & 3 & -1 & -1 +\end{array}\right]\sim\left[\begin{array}{cccc} +1 & 1 & 0 & 0\\ +0 & 0 & 1 & 1 +\end{array}\right]\Rightarrow x=-y,\quad z=-w\Rightarrow\tilde{u_{1}}=\left(1,-1,0,0\right),\quad\tilde{u_{2}}=\left(0,0,1,-1\right) +\] + +\end_inset + + +\begin_inset Formula +\[ +u_{1}=\tilde{u_{1}}/\left|\left|\tilde{u_{1}}\right|\right|=\left(\frac{1}{\sqrt{2}},\frac{-1}{\sqrt{2}},0,0\right)\quad\quad\quad u_{2}=\tilde{u_{2}}/\left|\left|\tilde{u}_{2}\right|\right|=\left(0,0,\frac{1}{\sqrt{2}},\frac{-1}{\sqrt{2}}\right) +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +y=\sum_{i=1}^{n}\frac{\left\langle y,v_{i}\right\rangle }{\left\langle v_{i},v_{i}\right\rangle }v_{i}=\left\langle \left(6,0,2,0\right),\left(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)\right\rangle v_{1}+\left\langle \left(6,0,2,0\right),\left(\frac{1}{2},\frac{1}{2},\frac{-1}{2},\frac{-1}{2}\right)\right\rangle v_{2}+ +\] + +\end_inset + + \begin_inset Formula \[ -y=\sum_{i=1}^{n}\frac{\left\langle y,v_{i}\right\rangle }{\left\langle v_{i},v_{i}\right\rangle }v_{i}=\left\langle \left(-6,0,2,0\right),\left(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)\right\rangle v_{i}+\left\langle \left(-6,0,2,0\right),\left(\frac{1}{2},\frac{1}{2},\frac{-1}{2},\frac{-1}{2}\right)\right\rangle \left(\frac{1}{2},\frac{1}{2},\frac{-1}{2},\frac{-1}{2}\right)= ++\left\langle \left(6,0,2,0\right),\left(\frac{1}{\sqrt{2}},\frac{-1}{\sqrt{2}},0,0\right)\right\rangle u_{1}+\left\langle \left(6,0,2,0\right),\left(0,0,\frac{1}{\sqrt{2}},\frac{-1}{\sqrt{2}},\right)\right\rangle u_{2}=4v_{1}+2v_{2}+\frac{6}{\sqrt{2}}u_{1}+\frac{2}{\sqrt{2}}u_{2}= \] \end_inset @@ -830,7 +883,7 @@ y=\sum_{i=1}^{n}\frac{\left\langle y,v_{i}\right\rangle }{\left\langle v_{i},v_{ \begin_inset Formula \[ -=4\left(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)+2\left(\frac{1}{2},\frac{1}{2},\frac{-1}{2},\frac{-1}{2}\right)=\left(2,2,2,2\right)+\left(1,1,-1,-1\right) +=\left(2,2,2,2\right)+\left(1,1,-1,-1\right)+\left(3,-3,0,0\right)+\left(0,0,1,-1\right)=\left(3,3,1,1\right)\in W+\left(3,-3,1,-1\right)\in W^{\perp} \] \end_inset @@ -1092,43 +1145,5 @@ Rokopisi, ki sledijo, naj služijo le kot dokaz samostojnega reševanja. Zavedam se namreč njihovega neličnega izgleda. \end_layout -\begin_layout Standard -\begin_inset External - template PDFPages - filename /mnt/slu/shramba/upload/www/d/1ladn8a.jpg - -\end_inset - - -\begin_inset External - template PDFPages - filename /mnt/slu/shramba/upload/www/d/1ladn8b.jpg - -\end_inset - - -\begin_inset External - template PDFPages - filename /mnt/slu/shramba/upload/www/d/1ladn8c.jpg - -\end_inset - - -\begin_inset External - template PDFPages - filename /mnt/slu/shramba/upload/www/d/3ladn8.jpg - -\end_inset - - -\begin_inset External - template PDFPages - filename /mnt/slu/shramba/upload/www/d/4ladn8.jpg - -\end_inset - - -\end_layout - \end_body \end_document |