https://fweb.wallawalla.edu/class-wiki/index.php?title=Orthogonal_functions&feed=atom&action=historyOrthogonal functions - Revision history2024-03-29T15:27:06ZRevision history for this page on the wikiMediaWiki 1.39.1https://fweb.wallawalla.edu/class-wiki/index.php?title=Orthogonal_functions&diff=11225&oldid=prevFrohro: /* Other resources on orthogonality */2014-09-29T03:15:09Z<p><span dir="auto"><span class="autocomment">Other resources on orthogonality</span></span></p>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[http://www.uvm.edu/~pdodds/files/papers/others/2000/sterian2000a.pdf Another professor's view of this topic...]</div></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[http://www.uvm.edu/~pdodds/files/papers/others/2000/sterian2000a.pdf Another professor's view of this topic...]</div></td>
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</table>Frohrohttps://fweb.wallawalla.edu/class-wiki/index.php?title=Orthogonal_functions&diff=11224&oldid=prevFrohro: /* Other resources on orthogonality */2014-09-29T03:14:52Z<p><span dir="auto"><span class="autocomment">Other resources on orthogonality</span></span></p>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Other resources on orthogonality==</div></td>
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</table>Frohrohttps://fweb.wallawalla.edu/class-wiki/index.php?title=Orthogonal_functions&diff=4454&oldid=prevFonggr: /* Can we write functions in an analogous way compared to the way we write vectors? */2008-11-07T08:16:46Z<p><span dir="auto"><span class="autocomment">Can we write functions in an analogous way compared to the way we write vectors?</span></span></p>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Remember we wrote <math> \vec \bold v = \sum_{k=1}^3 v_k \hat \bold a_k </math>. Can we write something similar for a function, f(t) defined for a t element of the reals? Well maybe.... If the sum over the dummy index k becomes an integral over the dummy variable, x, and the unit vectors <math> \vec \bold a_k </math> are replaced with something like <math> \delta(x-t) </math>, the [http://en.wikipedia.org/wiki/Delta_function Dirac delta function]. The result would look something like this:</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Remember we wrote <math> \vec \bold v = \sum_{k=1}^3 v_k \hat \bold a_k </math>. Can we write something similar for a function,<ins style="font-weight: bold; text-decoration: none;"> <math></ins> f(t)<ins style="font-weight: bold; text-decoration: none;">\,\! </math></ins> defined for a<ins style="font-weight: bold; text-decoration: none;"> <math></ins> t<ins style="font-weight: bold; text-decoration: none;"> \,\! </math></ins> element of the reals? Well maybe.... If the sum over the dummy index<ins style="font-weight: bold; text-decoration: none;"> <math></ins> k <ins style="font-weight: bold; text-decoration: none;">\,\!</math></ins>becomes an integral over the dummy variable,<ins style="font-weight: bold; text-decoration: none;"> <math></ins> x<ins style="font-weight: bold; text-decoration: none;"> \,\! </math></ins>, and the unit vectors <math> \vec \bold a_k </math> are replaced with something like <math> \delta(x-t) <ins style="font-weight: bold; text-decoration: none;">\,\!</ins></math>, the [http://en.wikipedia.org/wiki/Delta_function Dirac delta function]. The result would look something like this:</div></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math> f(t) = \int_{- \infty}^\infty f(x) \delta (x-t) dx </math>.</div></td>
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</table>Fonggrhttps://fweb.wallawalla.edu/class-wiki/index.php?title=Orthogonal_functions&diff=4453&oldid=prevFonggr: /* Functions and vectors, an analogy */2008-11-07T08:12:38Z<p><span dir="auto"><span class="autocomment">Functions and vectors, an analogy</span></span></p>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Functions and vectors, an analogy==</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>We may think of the number of the direction, <math> k \,\!</math>, as the independent variable of a vector and the component in that direction, <math> v_k \,\!</math> as the dependent variable of the vector <math> \vec \bold v </math> in a similar way to the way we think of t as the independent variable of a function <math> f() \,\! </math>, where <math> f(t)\,\! </math> is the dependent variable of <math> f\,\! </math>. Probably the biggest difference here is that t often takes on real values from <math> - \infty </math> to <math> \infty </math>, and <math> k \in {1, 2, 3} </math>. Using this analogy, we may think of a function as a vector having an uncountably infinite number of dimensions. </div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>We may think of the number of the direction, <math> k \,\!</math>, as the independent variable of a vector and the component in that direction, <math> v_k \,\!</math> as the dependent variable of the vector <math> \vec \bold v </math> in a similar way to the way we think of<ins style="font-weight: bold; text-decoration: none;"> <math></ins> t<ins style="font-weight: bold; text-decoration: none;"> \,\! </math></ins> as the independent variable of a function <math> f() \,\! </math>, where <math> f(t)\,\! </math> is the dependent variable of <math> f\,\! </math>. Probably the biggest difference here is that t often takes on real values from <math> - \infty </math> to <math> \infty </math>, and <math> k \in {1, 2, 3} </math>. Using this analogy, we may think of a function as a vector having an uncountably infinite number of dimensions. </div></td>
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</table>Fonggrhttps://fweb.wallawalla.edu/class-wiki/index.php?title=Orthogonal_functions&diff=4452&oldid=prevFonggr: /* Functions and vectors, an analogy */2008-11-07T08:12:19Z<p><span dir="auto"><span class="autocomment">Functions and vectors, an analogy</span></span></p>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Functions and vectors, an analogy==</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>We may think of the number of the direction, <math> k </math>, as the independent variable of a vector and the component in that direction, <math> v_k </math> as the dependent variable of the vector <math> \vec \bold v </math> in a similar way to the way we think of t as the independent variable of a function f(), where f(t) is the dependent variable of f. Probably the biggest difference here is that t often takes on real values from <math> - \infty </math> to <math> \infty </math>, and <math> k \in {1, 2, 3} </math>. Using this analogy, we may think of a function as a vector having an uncountably infinite number of dimensions. </div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>We may think of the number of the direction, <math> k <ins style="font-weight: bold; text-decoration: none;">\,\!</ins></math>, as the independent variable of a vector and the component in that direction, <math> v_k <ins style="font-weight: bold; text-decoration: none;">\,\!</ins></math> as the dependent variable of the vector <math> \vec \bold v </math> in a similar way to the way we think of t as the independent variable of a function<ins style="font-weight: bold; text-decoration: none;"> <math></ins> f()<ins style="font-weight: bold; text-decoration: none;"> \,\! </math></ins>, where<ins style="font-weight: bold; text-decoration: none;"> <math></ins> f(t)<ins style="font-weight: bold; text-decoration: none;">\,\! </math></ins> is the dependent variable of<ins style="font-weight: bold; text-decoration: none;"> <math></ins> f<ins style="font-weight: bold; text-decoration: none;">\,\! </math></ins>. Probably the biggest difference here is that t often takes on real values from <math> - \infty </math> to <math> \infty </math>, and <math> k \in {1, 2, 3} </math>. Using this analogy, we may think of a function as a vector having an uncountably infinite number of dimensions. </div></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>====Can we write functions in an analogous way compared to the way we write vectors?====</div></td>
<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>====Can we write functions in an analogous way compared to the way we write vectors?====</div></td>
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</table>Fonggrhttps://fweb.wallawalla.edu/class-wiki/index.php?title=Orthogonal_functions&diff=4451&oldid=prevFonggr: /* Inner products for functions */2008-11-07T08:11:18Z<p><span dir="auto"><span class="autocomment">Inner products for functions</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 01:11, 7 November 2008</td>
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<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===Inner products for functions===</div></td>
<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===Inner products for functions===</div></td>
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<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>[[Orthogonal functions#Inner products for vectors|Above]] we found that a vector inner product between <math>\vec \bold u </math> and <math>\vec \bold v </math> could be written as <math> \vec \bold u \<del style="font-weight: bold; text-decoration: none;">bullet </del>\vec \bold v = \sum_{k=1}^3 u_k v_k </math>. If we follow our above analogy, we should be able to replace the sum over k with an integral over x. There is one little notational problem, and that is we don't want to confuse the functional inner product with a simple muliply, so we need some new notation to denote this new inner product. In [http://en.wikipedia.org/wiki/Quantum_mechanics quantum mechanics], physicists use the [http://en.wikipedia.org/wiki/Bra-ket_notation bra-ket] notation. Let's borrow that.</div></td>
<td class="diff-marker" data-marker="+"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>[[Orthogonal functions#Inner products for vectors|Above]] we found that a vector inner product between <math>\vec \bold u </math> and <math>\vec \bold v </math> could be written as <math> \vec \bold u \<ins style="font-weight: bold; text-decoration: none;">cdot</ins>\vec \bold v = \sum_{k=1}^3 u_k v_k </math>. If we follow our above analogy, we should be able to replace the sum over k with an integral over x. There is one little notational problem, and that is we don't want to confuse the functional inner product with a simple muliply, so we need some new notation to denote this new inner product. In [http://en.wikipedia.org/wiki/Quantum_mechanics quantum mechanics], physicists use the [http://en.wikipedia.org/wiki/Bra-ket_notation bra-ket] notation. Let's borrow that.</div></td>
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<tr>
<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
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<tr>
<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math> <u|v> = \int_{-\infty}^\infty u^*(x) v(x) dx </math></div></td>
<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math> <u|v> = \int_{-\infty}^\infty u^*(x) v(x) dx </math></div></td>
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</table>Fonggrhttps://fweb.wallawalla.edu/class-wiki/index.php?title=Orthogonal_functions&diff=4449&oldid=prevFonggr: /* Normalization */2008-11-07T07:57:29Z<p><span dir="auto"><span class="autocomment">Normalization</span></span></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 00:57, 7 November 2008</td>
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<td colspan="2" class="diff-lineno">Line 23:</td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>When the <math> w_k = 1 \,\!</math> we have an orthonormal basis set, meaning that it is both orthogonal and that the basis vectors are normalized to unity (or have length one). Orthonormal vector systems are very popular. In fact they are the most common vector systems you will find. The reason they are so handy is each direction is uncoupled from the others.</div></td>
<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>When the <math> w_k = 1 \,\!</math> we have an orthonormal basis set, meaning that it is both orthogonal and that the basis vectors are normalized to unity (or have length one). Orthonormal vector systems are very popular. In fact they are the most common vector systems you will find. The reason they are so handy is each direction is uncoupled from the others.</div></td>
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<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
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<tr>
<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>For example, to find <math> v_n </math>, we take the inner product of the vector <math> \vec \bold v </math> with a unit vector in the nth direction, <math> \vec \bold a_n </math>. We write this operation like this: </div></td>
<td class="diff-marker" data-marker="+"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>For example, to find <math> v_n <ins style="font-weight: bold; text-decoration: none;">\,\!</ins></math>, we take the inner product of the vector <math> \vec \bold v </math> with a unit vector in the nth direction, <math> \vec \bold a_n </math>. We write this operation like this: </div></td>
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<tr>
<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
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<tr>
<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math> \vec \bold v \cdot \vec \bold a_n = \sum_{k=1}^3 v_k \vec \bold a_k \cdot \vec \bold a_n = \sum_{k=1}^3 v_k \delta_{k,n} = v_n </math></div></td>
<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math> \vec \bold v \cdot \vec \bold a_n = \sum_{k=1}^3 v_k \vec \bold a_k \cdot \vec \bold a_n = \sum_{k=1}^3 v_k \delta_{k,n} = v_n </math></div></td>
</tr>
</table>Fonggrhttps://fweb.wallawalla.edu/class-wiki/index.php?title=Orthogonal_functions&diff=4448&oldid=prevFonggr: /* Normalization */2008-11-07T07:56:18Z<p><span dir="auto"><span class="autocomment">Normalization</span></span></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 00:56, 7 November 2008</td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>where the <math> w_k \,\!</math> is the square of the length of <math> \vec \bold a_k </math> and the symbol <math> \delta_{k,n} \,\!</math>, known as the [http://en.wikipedia.org/wiki/Kronecker_delta Kronecker delta], is one when k = n and zero otherwise.</div></td>
<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>where the <math> w_k \,\!</math> is the square of the length of <math> \vec \bold a_k </math> and the symbol <math> \delta_{k,n} \,\!</math>, known as the [http://en.wikipedia.org/wiki/Kronecker_delta Kronecker delta], is one when k = n and zero otherwise.</div></td>
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<tr>
<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>=====Normalization=====</div></td>
<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>=====Normalization=====</div></td>
</tr>
<tr>
<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>When the <math> w_k = 1</math> we have an orthonormal basis set, meaning that it is both orthogonal and that the basis vectors are normalized to unity (or have length one). Orthonormal vector systems are very popular. In fact they are the most common vector systems you will find. The reason they are so handy is each direction is uncoupled from the others.</div></td>
<td class="diff-marker" data-marker="+"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>When the <math> w_k = 1<ins style="font-weight: bold; text-decoration: none;"> \,\!</ins></math> we have an orthonormal basis set, meaning that it is both orthogonal and that the basis vectors are normalized to unity (or have length one). Orthonormal vector systems are very popular. In fact they are the most common vector systems you will find. The reason they are so handy is each direction is uncoupled from the others.</div></td>
</tr>
<tr>
<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
</tr>
<tr>
<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>For example, to find <math> v_n </math>, we take the inner product of the vector <math> \vec \bold v </math> with a unit vector in the nth direction, <math> \vec \bold a_n </math>. We write this operation like this: </div></td>
<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>For example, to find <math> v_n </math>, we take the inner product of the vector <math> \vec \bold v </math> with a unit vector in the nth direction, <math> \vec \bold a_n </math>. We write this operation like this: </div></td>
</tr>
</table>Fonggrhttps://fweb.wallawalla.edu/class-wiki/index.php?title=Orthogonal_functions&diff=4447&oldid=prevFonggr: /* Orthogonality for vectors */2008-11-07T07:55:19Z<p><span dir="auto"><span class="autocomment">Orthogonality for vectors</span></span></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<col class="diff-marker" />
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<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 00:55, 7 November 2008</td>
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<td colspan="2" class="diff-lineno">Line 19:</td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>\vec \bold a_k \cdot \vec \bold a_n = w_k \delta_{k,n} </math></div></td>
<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>\vec \bold a_k \cdot \vec \bold a_n = w_k \delta_{k,n} </math></div></td>
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<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
</tr>
<tr>
<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>where the <math> w_k </math> is the square of the length of <math> \vec \bold a_k </math> and the symbol <math> \delta_{k,n} \,\!</math>, known as the [http://en.wikipedia.org/wiki/Kronecker_delta Kronecker delta], is one when k = n and zero otherwise.</div></td>
<td class="diff-marker" data-marker="+"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>where the <math> w_k <ins style="font-weight: bold; text-decoration: none;">\,\!</ins></math> is the square of the length of <math> \vec \bold a_k </math> and the symbol <math> \delta_{k,n} \,\!</math>, known as the [http://en.wikipedia.org/wiki/Kronecker_delta Kronecker delta], is one when k = n and zero otherwise.</div></td>
</tr>
<tr>
<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>=====Normalization=====</div></td>
<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>=====Normalization=====</div></td>
</tr>
<tr>
<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>When the <math> w_k = 1</math> we have an orthonormal basis set, meaning that it is both orthogonal and that the basis vectors are normalized to unity (or have length one). Orthonormal vector systems are very popular. In fact they are the most common vector systems you will find. The reason they are so handy is each direction is uncoupled from the others.</div></td>
<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>When the <math> w_k = 1</math> we have an orthonormal basis set, meaning that it is both orthogonal and that the basis vectors are normalized to unity (or have length one). Orthonormal vector systems are very popular. In fact they are the most common vector systems you will find. The reason they are so handy is each direction is uncoupled from the others.</div></td>
</tr>
</table>Fonggrhttps://fweb.wallawalla.edu/class-wiki/index.php?title=Orthogonal_functions&diff=4445&oldid=prevFonggr: /* Vector notation */2008-11-07T07:52:37Z<p><span dir="auto"><span class="autocomment">Vector notation</span></span></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
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<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 00:52, 7 November 2008</td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math> \vec \bold v = <1, 4, 3> </math> means that the vector <math> \vec \bold v </math> is one unit in the first direction (often the x direction), four units in the second direction (often the y direction), and three units in the last direction (often the z direction). We say the component of <math> \vec \bold v </math> in the second direction is 4. This is often written as <math> v_y = 4 \,\!</math>.</div></td>
<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math> \vec \bold v = <1, 4, 3> </math> means that the vector <math> \vec \bold v </math> is one unit in the first direction (often the x direction), four units in the second direction (often the y direction), and three units in the last direction (often the z direction). We say the component of <math> \vec \bold v </math> in the second direction is 4. This is often written as <math> v_y = 4 \,\!</math>.</div></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>====Vector notation====</div></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>====Vector notation====</div></td>
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<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>We don't have to use x, y, and z as the direction names; we can use numbers, like 1, 2, and 3 instead. The advantage of this is that it leads to more compact notation, and extends to more than three dimensions much better. For example we could say <math> v_2 = 4 \,\!</math> instead of <math> v_y = 4 \,\!</math>. Instead of writing <math> \vec \bold v = <1, 4, 3> </math> we can write <math> \vec \bold v = \sum_{k=1}^3 v_k \hat \bold a_k </math> where the <math>\hat \bold a_k </math> denotes a basis vector in the kth direction, <math>v_1 = 1,</math> <math> v_2 = 4, </math> and <math> v_3 = 3</math>. The idea of basis vectors was implicit in the notation <math> \vec \bold v = <1, 4, 3> </math>.</div></td>
<td class="diff-marker" data-marker="+"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>We don't have to use x, y, and z as the direction names; we can use numbers, like 1, 2, and 3 instead. The advantage of this is that it leads to more compact notation, and extends to more than three dimensions much better. For example we could say <math> v_2 = 4 \,\!</math> instead of <math> v_y = 4 \,\!</math>. Instead of writing <math> \vec \bold v = <1, 4, 3> </math> we can write <math> \vec \bold v = \sum_{k=1}^3 v_k \hat \bold a_k </math> where the <math>\hat \bold a_k </math> denotes a basis vector in the kth direction, <math>v_1 = 1,<ins style="font-weight: bold; text-decoration: none;"> \,\!</ins></math> <math> v_2 = 4, <ins style="font-weight: bold; text-decoration: none;">\,\!</ins></math> and <math> v_3 = 3<ins style="font-weight: bold; text-decoration: none;"> \,\!</ins></math>. The idea of basis vectors was implicit in the notation <math> \vec \bold v = <1, 4, 3> </math>.</div></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===Inner products for vectors===</div></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===Inner products for vectors===</div></td>
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</table>Fonggr