https://artofproblemsolving.com/wiki/index.php?title=Cauchy-schwarz_inequality&feed=atom&action=history
Cauchy-schwarz inequality - Revision history
2024-03-28T16:31:47Z
Revision history for this page on the wiki
MediaWiki 1.31.1
https://artofproblemsolving.com/wiki/index.php?title=Cauchy-schwarz_inequality&diff=2808&oldid=prev
Dts: If you say so.
2006-06-18T21:22:34Z
<p>If you say so.</p>
<table class="diff diff-contentalign-left" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 21:22, 18 June 2006</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1" >Line 1:</td>
<td colspan="2" class="diff-lineno">Line 1:</td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; 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><del class="diffchange diffchange-inline">Consider the [[quadratic]], </del></div></td><td class='diff-marker'>+</td><td style="color: #222; 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><ins class="diffchange diffchange-inline">#REDIRECT </ins>[[Cauchy-Schwarz Inequality]]</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; 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><del class="diffchange diffchange-inline"><math>(a_1x+b_1)^2+(a_2x+b_2)^2+...(a_nx+b_n)^2 = 0</math>.</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; 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><del class="diffchange diffchange-inline">Expanding, we find the equation to be of the form <math>Ax^2+Bx+C</math>, where <math>A=\sum_{i=1}^n a_i^2</math>, <math>B=2\sum_{j=1}^n a_jb_j</math>, and <math>C=\sum_{k=1}^n b_k^2.</math></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; 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><del class="diffchange diffchange-inline">Since the equation is always greater than or equal to 0, <math>B^2-4AC \leq 0</math>.  Substituting the above values of A, B, and C leaves us with the '''Cauchy-Schwarz Inequality''', which states that</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; 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><del class="diffchange diffchange-inline"><math>(a_1b_1+a_2b_2+...+a_nb_n)^2 \leq (a_1^2+a_2^2+...+a_n^2)(b_1^2+b_2^2+...+b_n^2)</math>,</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; 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><del class="diffchange diffchange-inline">or, in the more compact [[sigma notation]], </del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; 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><del class="diffchange diffchange-inline"><math>\left(\sum a_ib_i\right) \leq \left(\sum a_i^2\right)\left(\sum b_i^2\right)</math></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; 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><del class="diffchange diffchange-inline">''This page should be deleted as it has become obsolete with the more complete </del>[[Cauchy-Schwarz Inequality]]<del class="diffchange diffchange-inline">'' page with a capital S in shwarz"</del></div></td><td colspan="2"> </td></tr>
</table>
Dts
https://artofproblemsolving.com/wiki/index.php?title=Cauchy-schwarz_inequality&diff=2543&oldid=prev
Quantum leap at 05:12, 18 June 2006
2006-06-18T05:12:16Z
<p></p>
<table class="diff diff-contentalign-left" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 05:12, 18 June 2006</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l6" >Line 6:</td>
<td colspan="2" class="diff-lineno">Line 6:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; 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>or, in the more compact [[sigma notation]],  </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; 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>or, in the more compact [[sigma notation]],  </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; 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>\left(\sum a_ib_i\right) \leq \left(\sum a_i^2\right)\left(\sum b_i^2\right)</math></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; 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>\left(\sum a_ib_i\right) \leq \left(\sum a_i^2\right)\left(\sum b_i^2\right)</math></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; 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><ins style="font-weight: bold; text-decoration: none;">''This page should be deleted as it has become obsolete with the more complete [[Cauchy-Schwarz Inequality]]'' page with a capital S in shwarz"</ins></div></td></tr>
</table>
Quantum leap
https://artofproblemsolving.com/wiki/index.php?title=Cauchy-schwarz_inequality&diff=2539&oldid=prev
Quantum leap at 04:55, 18 June 2006
2006-06-18T04:55:10Z
<p></p>
<table class="diff diff-contentalign-left" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 04:55, 18 June 2006</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l2" >Line 2:</td>
<td colspan="2" class="diff-lineno">Line 2:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; 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>(a_1x+b_1)^2+(a_2x+b_2)^2+...(a_nx+b_n)^2 = 0</math>.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; 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>(a_1x+b_1)^2+(a_2x+b_2)^2+...(a_nx+b_n)^2 = 0</math>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; 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>Expanding, we find the equation to be of the form <math>Ax^2+Bx+C</math>, where <math>A=\sum_{i=1}^n a_i^2</math>, <math>B=2\sum_{j=1}^n a_jb_j</math>, and <math>C=\sum_{k=1}^n b_k^2.</math></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; 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>Expanding, we find the equation to be of the form <math>Ax^2+Bx+C</math>, where <math>A=\sum_{i=1}^n a_i^2</math>, <math>B=2\sum_{j=1}^n a_jb_j</math>, and <math>C=\sum_{k=1}^n b_k^2.</math></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; 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><del class="diffchange diffchange-inline">The </del>equation <del class="diffchange diffchange-inline">will have a solution when the discriminant </del>is greater than or equal to 0, <del class="diffchange diffchange-inline">so </del><math>B^2-4AC \<del class="diffchange diffchange-inline">geq </del>0</math>.  Substituting the above values of A, B, and C leaves us with the '''Cauchy-Schwarz Inequality''', which states that</div></td><td class='diff-marker'>+</td><td style="color: #222; 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><ins class="diffchange diffchange-inline">Since the </ins>equation is <ins class="diffchange diffchange-inline">always </ins>greater than or equal to 0, <math>B^2-4AC \<ins class="diffchange diffchange-inline">leq </ins>0</math>.  Substituting the above values of A, B, and C leaves us with the '''Cauchy-Schwarz Inequality''', which states that</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; 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><math>(a_1b_1+a_2b_2+...+a_nb_n)^2 \<del class="diffchange diffchange-inline">geq </del>(a_1^2+a_2^2+...+a_n^2)(b_1^2+b_2^2+...+b_n^2)</math>,</div></td><td class='diff-marker'>+</td><td style="color: #222; 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><math>(a_1b_1+a_2b_2+...+a_nb_n)^2 \<ins class="diffchange diffchange-inline">leq </ins>(a_1^2+a_2^2+...+a_n^2)(b_1^2+b_2^2+...+b_n^2)</math>,</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; 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>or, in the more compact [[sigma notation]],  </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; 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>or, in the more compact [[sigma notation]],  </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; 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><math>\left(\sum a_ib_i\right) \<del class="diffchange diffchange-inline">geq </del>\left(\sum a_i^2\right)\left(\sum b_i^2\right)</math></div></td><td class='diff-marker'>+</td><td style="color: #222; 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><math>\left(\sum a_ib_i\right) \<ins class="diffchange diffchange-inline">leq </ins>\left(\sum a_i^2\right)\left(\sum b_i^2\right)</math></div></td></tr>
</table>
Quantum leap
https://artofproblemsolving.com/wiki/index.php?title=Cauchy-schwarz_inequality&diff=2524&oldid=prev
Quantum leap at 04:19, 18 June 2006
2006-06-18T04:19:28Z
<p></p>
<p><b>New page</b></p><div>Consider the [[quadratic]], <br />
<math>(a_1x+b_1)^2+(a_2x+b_2)^2+...(a_nx+b_n)^2 = 0</math>.<br />
Expanding, we find the equation to be of the form <math>Ax^2+Bx+C</math>, where <math>A=\sum_{i=1}^n a_i^2</math>, <math>B=2\sum_{j=1}^n a_jb_j</math>, and <math>C=\sum_{k=1}^n b_k^2.</math><br />
The equation will have a solution when the discriminant is greater than or equal to 0, so <math>B^2-4AC \geq 0</math>. Substituting the above values of A, B, and C leaves us with the '''Cauchy-Schwarz Inequality''', which states that<br />
<math>(a_1b_1+a_2b_2+...+a_nb_n)^2 \geq (a_1^2+a_2^2+...+a_n^2)(b_1^2+b_2^2+...+b_n^2)</math>,<br />
or, in the more compact [[sigma notation]], <br />
<math>\left(\sum a_ib_i\right) \geq \left(\sum a_i^2\right)\left(\sum b_i^2\right)</math></div>
Quantum leap