https://enciklopedija.cc/index.php?action=history&feed=atom&title=Gram-Schmidtov_postupak 2026-07-15T20:07:38Z Povijest promjena ove stranice na wikiju MediaWiki 1.42.3 https://enciklopedija.cc/index.php?title=Gram-Schmidtov_postupak&diff=50535&oldid=prev 2021-08-23T05:07:39Z

<p>Bot: Automatski unos stranica</p> <p><b>Nova stranica</b></p><div>&lt;!--&#039;&#039;&#039;Gram-Schmidtov postupak&#039;&#039;&#039;--&gt;&#039;&#039;&#039;Gram-Schmidtov postupak&#039;&#039;&#039; je metoda u [[linearna algebra|linearnoj algebri]] koja služi za ortogonalizaciju skupa vektora u zadanom euklidskom prostoru.<br /> <br /> Postupak je sljedeći. Uzmimo na primjer [[vektorski prostor]] proizvoljne [[Dimenzija vektorskog prostora|dimenzije]] &#039;&#039;&#039;R&#039;&#039;&#039;&lt;sup&gt;&#039;&#039;n&#039;&#039;&lt;/sup&gt; baze {&#039;&#039;&#039;v&#039;&#039;&#039;&lt;sub&gt;1&lt;/sub&gt;, &#039;&#039;&#039;v&#039;&#039;&#039;&lt;sub&gt;2&lt;/sub&gt;, ... ,&#039;&#039;&#039;v&#039;&#039;&#039;&lt;sub&gt;n&lt;/sub&gt;}, Gram-Schmidtovim postupkom ortogonalizacije možemo transformirati bazu {&#039;&#039;&#039;v&#039;&#039;&#039;&lt;sub&gt;i&lt;/sub&gt;} u ortonormiranu bazu, {&#039;&#039;&#039;u&#039;&#039;&#039;&lt;sub&gt;i&lt;/sub&gt;}. Prvo normaliziramo &#039;&#039;&#039;v&#039;&#039;&#039;&lt;sub&gt;1&lt;/sub&gt;: &#039;&#039;&#039;u&#039;&#039;&#039;&lt;sub&gt;1&lt;/sub&gt;=&#039;&#039;&#039;v&#039;&#039;&#039;&lt;sub&gt;1&lt;/sub&gt;/||&#039;&#039;&#039;v&#039;&#039;&#039;&lt;sub&gt;1&lt;/sub&gt;||.<br /> <br /> Nakon toga izračunavamo &#039;&#039;&#039;w&#039;&#039;&#039;&lt;sub&gt;2&lt;/sub&gt;=&#039;&#039;&#039;v&#039;&#039;&#039;&lt;sub&gt;2&lt;/sub&gt;-&lt;&#039;&#039;&#039;v&#039;&#039;&#039;&lt;sub&gt;2&lt;/sub&gt;,&#039;&#039;&#039;u&#039;&#039;&#039;&lt;sub&gt;1&lt;/sub&gt;&gt;&#039;&#039;&#039;u&#039;&#039;&#039;&lt;sub&gt;1&lt;/sub&gt;, pa normaliziramo &#039;&#039;&#039;w&#039;&#039;&#039;&lt;sub&gt;2&lt;/sub&gt;: &#039;&#039;&#039;u&#039;&#039;&#039;&lt;sub&gt;2&lt;/sub&gt;=&#039;&#039;&#039;w&#039;&#039;&#039;&lt;sub&gt;2&lt;/sub&gt;/||&#039;&#039;&#039;w&#039;&#039;&#039;&lt;sub&gt;2&lt;/sub&gt;||<br /> <br /> Ovaj postupak primjenimo za sve vektore iz baze {&#039;&#039;&#039;v&#039;&#039;&#039;&lt;sub&gt;i&lt;/sub&gt;}: &#039;&#039;&#039;w&#039;&#039;&#039;&lt;sub&gt;i+1&lt;/sub&gt;=&#039;&#039;&#039;v&#039;&#039;&#039;&lt;sub&gt;i+1&lt;/sub&gt;-&lt;&#039;&#039;&#039;v&#039;&#039;&#039;&lt;sub&gt;i+1&lt;/sub&gt;,&#039;&#039;&#039;u&#039;&#039;&#039;&lt;sub&gt;i&lt;/sub&gt;ui&gt;- ... - &lt;&#039;&#039;&#039;v&#039;&#039;&#039;&lt;sub&gt;i+1&lt;/sub&gt;,&#039;&#039;&#039;u&#039;&#039;&#039;&lt;sub&gt;1&lt;/sub&gt;&gt;&#039;&#039;&#039;u&#039;&#039;&#039;&lt;sub&gt;1&lt;/sub&gt; i &#039;&#039;&#039;u&#039;&#039;&#039;&lt;sub&gt;i+1&lt;/sub&gt;=&#039;&#039;&#039;w&#039;&#039;&#039;&lt;sub&gt;i+1&lt;/sub&gt;/||&#039;&#039;&#039;w&#039;&#039;&#039;&lt;sub&gt;i+1&lt;/sub&gt;||. Vektori {&#039;&#039;&#039;u&#039;&#039;&#039;&lt;sub&gt;1&lt;/sub&gt;, ... ,&#039;&#039;&#039;v&#039;&#039;&#039;&lt;sub&gt;n&lt;/sub&gt;} su linearno nezavisni, i stoga čine bazu vektorskog prostora &#039;&#039;&#039;R&#039;&#039;&#039;&lt;sup&gt;&#039;&#039;n&#039;&#039;&lt;/sup&gt;.<br /> <br /> == Primjer ==<br /> <br /> Uzmimo sljedeći skup vektora u &#039;&#039;&#039;R&#039;&#039;&#039;&lt;sup&gt;&#039;&#039;n&#039;&#039;&lt;/sup&gt; (sa uobičajenim skalarnim produktom)<br /> :&lt;math&gt;S = \left\lbrace\mathbf{v}_1=\begin{pmatrix} 3 \\ 1\end{pmatrix}, \mathbf{v}_2=\begin{pmatrix}2 \\2\end{pmatrix}\right\rbrace.&lt;/math&gt;<br /> <br /> Sad primjenimo Gram-Schmidtov postupak kako bismo dobili ortogonalni skup vektora:<br /> :&lt;math&gt;\mathbf{u}_1=\mathbf{v}_1=\begin{pmatrix}3\\1\end{pmatrix}&lt;/math&gt;<br /> :&lt;math&gt; \mathbf{u}_2 = \mathbf{v}_2 - \mathrm{proj}_{\mathbf{u}_1} \, \mathbf{v}_2 = \begin{pmatrix}2\\2\end{pmatrix} - \mathrm{proj}_{({3 \atop 1})} \, {\begin{pmatrix}2\\2\end{pmatrix}} = \begin{pmatrix} -2/5 \\6/5 \end{pmatrix}. &lt;/math&gt;<br /> <br /> Provjerimo vektore &#039;&#039;&#039;u&#039;&#039;&#039;&lt;sub&gt;1&lt;/sub&gt; i &#039;&#039;&#039;u&#039;&#039;&#039;&lt;sub&gt;2&lt;/sub&gt; kako bismo utvrdili da su zaista ortogonalni:<br /> :&lt;math&gt;\langle\mathbf{u}_1,\mathbf{u}_2\rangle = \left\langle \begin{pmatrix}3\\1\end{pmatrix}, \begin{pmatrix}-2/5\\6/5\end{pmatrix} \right\rangle = -\frac65 + \frac65 = 0.&lt;/math&gt;<br /> <br /> Sada ih možemo normalizirati, tako što ćemo ih podijeliti s njihovim duljinama:<br /> <br /> [[Datoteka:Gram–Schmidt process.svg|desno|mini|Prvi koraci Gram-Schmidtovog postupka.]]<br /> <br /> :&lt;math&gt;\mathbf{e}_1 = {1 \over \sqrt {10}}\begin{pmatrix}3\\1\end{pmatrix}&lt;/math&gt;<br /> :&lt;math&gt;\mathbf{e}_2 = {1 \over \sqrt{40 \over 25}} \begin{pmatrix}-2/5\\6/5\end{pmatrix}<br /> = {1\over\sqrt{10}} \begin{pmatrix}-1\\3\end{pmatrix}. &lt;/math&gt;<br /> <br /> [[Kategorija:Linearna algebra]]</div>

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