https://drorbn.net/index.php?action=history&feed=atom&title=09-240%2FClassnotes_for_Tuesday_October_13 2026-05-04T21:39:16Z Revision history for this page on the wiki MediaWiki 1.39.6 https://drorbn.net/index.php?title=09-240/Classnotes_for_Tuesday_October_13&diff=8833&oldid=prev 2009-12-07T07:26:27Z

<p>class notes warning</p> <table style="background-color: #fff; color: #202122;" 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: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 03:26, 7 December 2009</td> </tr><tr> <td colspan="2" class="diff-lineno">Line 1:</td> <td colspan="2" class="diff-lineno">Line 1:</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>{{09-240/Navigation}}</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>{{09-240/Navigation}}</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>{{09-240/Class Notes Warning}}</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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;"><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>&lt;gallery&gt;</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>&lt;gallery&gt;</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>Image:Oct 13 notes page 1.JPG|[[User:Bright|Bright]] - 1</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>Image:Oct 13 notes page 1.JPG|[[User:Bright|Bright]] - 1</div></td> </tr> </table>

C8sd https://drorbn.net/index.php?title=09-240/Classnotes_for_Tuesday_October_13&diff=8828&oldid=prev 2009-12-07T06:39:14Z

<p>Add partial notes.</p> <table style="background-color: #fff; color: #202122;" 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: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 02:39, 7 December 2009</td> </tr><tr> <td colspan="2" class="diff-lineno">Line 1:</td> <td colspan="2" class="diff-lineno">Line 1:</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>{{09-240/Navigation}}</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>{{09-240/Navigation}}</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>&lt;gallery&gt;</div></td> </tr> <tr> <td class="diff-marker"><a class="mw-diff-movedpara-left" title="Paragraph was moved. Click to jump to new location." href="#movedpara_4_0_rhs">&#x26AB;</a></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><a name="movedpara_2_0_lhs"></a><del style="font-weight: bold; text-decoration: none;">[[</del>Image:Oct 13 notes page <del style="font-weight: bold; text-decoration: none;">1</del>.JPG|<del style="font-weight: bold; text-decoration: none;">600px]]</del></div></td> <td colspan="2" class="diff-empty diff-side-added"></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><del style="font-weight: bold; text-decoration: none;">[[</del>Image:Oct 13 notes page <del style="font-weight: bold; text-decoration: none;">2</del>.JPG|<del style="font-weight: bold; text-decoration: none;">600px</del>]]</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>Image:Oct 13 notes page <ins style="font-weight: bold; text-decoration: none;">1</ins>.JPG|<ins style="font-weight: bold; text-decoration: none;">[[User:Bright|Bright</ins>]]<ins style="font-weight: bold; text-decoration: none;"> - 1</ins></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><del style="font-weight: bold; text-decoration: none;">[[</del>Image:Oct 13 notes page <del style="font-weight: bold; text-decoration: none;">3</del>.JPG|<del style="font-weight: bold; text-decoration: none;">600px]]</del></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>Image:Oct 13 notes page <ins style="font-weight: bold; text-decoration: none;">2</ins>.JPG|<ins style="font-weight: bold; text-decoration: none;">2</ins></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><del style="font-weight: bold; text-decoration: none;">[[</del>Image:Oct 13 notes page <del style="font-weight: bold; text-decoration: none;">4</del>.JPG|<del style="font-weight: bold; text-decoration: none;">600px]]</del></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>Image:Oct 13 notes page <ins style="font-weight: bold; text-decoration: none;">3</ins>.JPG|<ins style="font-weight: bold; text-decoration: none;">3</ins></div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker"><a class="mw-diff-movedpara-right" title="Paragraph was moved. Click to jump to old location." href="#movedpara_2_0_lhs">&#x26AB;</a></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><a name="movedpara_4_0_rhs"></a>Image:Oct 13 notes page <ins style="font-weight: bold; text-decoration: none;">4</ins>.JPG|<ins style="font-weight: bold; text-decoration: none;">4</ins></div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>&lt;/gallery&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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;"><br /></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>== Replacement Theorem ==</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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;"><br /></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>...</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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;"><br /></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>&lt;ol&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>&lt;li value="3"&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>dim('''V''') = ''n''</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>&lt;ol style="list-style-type: lower-alpha"&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>&lt;li&gt;If ''G'' generates '''V''' then &lt;math&gt;|G| \ge n&lt;/math&gt;. If also &lt;math&gt;\,\! |G| = n&lt;/math&gt; then ''G'' is a basis.&lt;/li&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>&lt;li&gt;If ''L'' is linearly dependent then &lt;math&gt;|L| \le n&lt;/math&gt;. If also &lt;math&gt;\,\! |L| = n&lt;/math&gt; then ''L'' is a basis. If also &lt;math&gt;\,\! |L| &lt; n&lt;/math&gt; then ''L'' can be extended to a basis.</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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;"><br /></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>&lt;u&gt;Proofs&lt;/u&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>''a1.'' If ''G'' has a subset which is a basis then that subset has ''n'' elements, so &lt;math&gt;|G| \ge n&lt;/math&gt;. &lt;br /&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>''a2.'' Let &lt;math&gt;\beta&lt;/math&gt; be a basis of '''V''', then &lt;math&gt;\,\! |B| = n&lt;/math&gt;. Now use replacement with ''G'' &amp; ''L'' = &lt;math&gt;\beta&lt;/math&gt;. Hence, &lt;math&gt;|G| \ge |L| = |B| = n&lt;/math&gt;.&lt;br /&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>&lt;u&gt;a.&lt;/u&gt; From a1 and a2, we know &lt;math&gt;|G| \ge n&lt;/math&gt;. If &lt;math&gt;\,\! |G| = n&lt;/math&gt; then ''G'' contains a basis &lt;math&gt;\beta&lt;/math&gt;. But &lt;math&gt;\,\! |B| = n&lt;/math&gt;, so &lt;math&gt;\,\! |G| = \beta&lt;/math&gt;, and hence ''G'' is a basis.&lt;br /&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>&lt;br /&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>&lt;u&gt;b.&lt;/u&gt; Use replacement with ''G' being some basis of '''V'''. |''G''| = ''n''.&lt;br /&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>&lt;br /&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>If &lt;math&gt;|L| = |G|&lt;/math&gt; then &lt;math&gt;|R| = n = |G|&lt;/math&gt;, so &lt;math&gt;R = G&lt;/math&gt;, so &lt;math&gt;(G \backslash R) \cup L&lt;/math&gt; generates, so ''L'' generates, so ''L'' is a basis since it is linearly independent.&lt;br /&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>&lt;br /&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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 have that ''L'' is basis. If &lt;math&gt;|L| &lt; |G|&lt;/math&gt; the nagain find &lt;math&gt;R \subset G&lt;/math&gt; such that &lt;math&gt;|R| = |L|&lt;/math&gt; and &lt;math&gt;(G \backslash R) \cup L&lt;/math&gt; generates.&lt;br /&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>&lt;br /&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>1. &lt;math&gt;\beta&lt;/math&gt; generates '''V'''. &lt;br /&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>2. &lt;math&gt;|B| \le |G| - |R| + |L| = n&lt;/math&gt;. So by part a, &lt;math&gt;\beta&lt;/math&gt; is a basis.&lt;br /&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>3. </div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>&lt;/li&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>&lt;/ol&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>&lt;/li&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>&lt;li value="4"&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>If '''V''' is finite-dimensional (f.d.) and &lt;math&gt;\mathbf W \subset \mathbf V&lt;/math&gt; is a subspace of '''V''', then '''W''' is also finite, and &lt;math&gt;\operatorname{dim}(\mathbf W) \le \operatorname{dim}(\mathbf V)&lt;/math&gt;. If also dim('''W''') = dim('''V''') then '''W''' = '''V''' and if dim('''W''') &lt; dim('''V''') then any basis of '''W''' can be extended to a basis of '''V'''.&lt;br /&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>&lt;br /&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>Proof: Assuming '''W''' is finite-dimensional, pick a basis &lt;math&gt;\beta&lt;/math&gt; of '''W'''; &lt;math&gt;\beta&lt;/math&gt; is linearly independent in '''V''' so by Corollary 3 of part b, &lt;math&gt;|B| \le \operatorname{dim}(\mathbf V) \Rightarrow \operatorname{dim}(\mathbf W) = |B| \le \operatorname{dim}(\mathbf V)&lt;/math&gt;. So span(&lt;math&gt;\beta&lt;/math&gt;) = '''V''' = '''W''' so '''V''' = '''W'''. ...&lt;br /&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>&lt;br /&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>Assume '''W''' is not finite-dimensional. &lt;math&gt;\mathbf W \ne \{0\}&lt;/math&gt; so pick a &lt;math&gt;x_1 \in W&lt;/math&gt; such that &lt;math&gt;x_1 \ne 0&lt;/math&gt;. So &lt;math&gt;\{x_1\}&lt;/math&gt; is linearly independent in '''W''', and &lt;math&gt;\operatorname{span}(\{x_1\}) \subsetneq \mathbf W&lt;/math&gt;. Pick &lt;math&gt;x_2 \in W \backslash \operatorname{span}(\{x_1\})&lt;/math&gt;. So &lt;math&gt;\{x_1, x_2\}&lt;/math&gt; is linearly dependent and &lt;math&gt;\operatorname{span}(\{x_1, x_2\}) \subsetneq \mathbf W&lt;/math&gt;. Pick &lt;math&gt;x_3 \in W \backslash \operatorname{span}(\{x_1, x_2\})&lt;/math&gt; ... continue in this way to get a sequence &lt;math&gt;x_1, x_2, \ldots, x_{n+1}&lt;/math&gt; where ''n'' = dim('''V''') and &lt;math&gt;\{x_1, \ldots, x_{n+1}\}&lt;/math&gt; is linearly independent. There is a contradiction by Corollary 3.b.</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>&lt;/li&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>&lt;/ol&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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;"><br /></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>== The Lagrange Interpolation Formula ==</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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;"><br /></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>[Aside: &lt;math&gt;(a - x)(b - x) \ldots (z - x) = 0&lt;/math&gt; because &lt;math&gt;x - x = 0&lt;/math&gt;]</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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;"><br /></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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 &lt;math&gt;1 \le i &lt; n,&lt;/math&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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;"><br /></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>Let &lt;math&gt;x_i&lt;/math&gt; be distinct points in &lt;math&gt;\real&lt;/math&gt;.</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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;"><br /></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>Let &lt;math&gt;y_i&lt;/math&gt; be any points in &lt;math&gt;\real&lt;/math&gt;.</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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;"><br /></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>Can you find a polynomial &lt;math&gt;P \in \mathbb P_n(\real)&lt;/math&gt; such that &lt;math&gt;P(x_i) = y_i&lt;/math&gt;? Is it unique?</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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;"><br /></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>'''Example''':</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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;"><br /></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>: &lt;math&gt;x_i = 0, 1, 3&lt;/math&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>: &lt;math&gt;y_i = 5, 2, 2&lt;/math&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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;"><br /></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>Can we find a &lt;math&gt;P \in \mathbb P_2&lt;/math&gt; such that</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>: &lt;math&gt;P(0) = 5&lt;/math&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>: &lt;math&gt;P(1) = 2&lt;/math&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>: &lt;math&gt;P(2) = 2&lt;/math&gt;?</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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;"><br /></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>Solution: Let &lt;math&gt;\tilde P_i(x) = \prod_{j=1, j \ne 1}^{n+1} (x - x_j) \in \mathbb P_n(\real)&lt;/math&gt;. (Remember [http://en.wikipedia.org/wiki/Capital_pi_notation capital pi notation].)</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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;"><br /></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>Then &lt;math&gt;\tilde P_i(x_k) = \begin{cases}</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>0 &amp; i \ne k \\</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>\prod_{i \ne j} (x_i - x_j) &amp; i = k \\</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>\end{cases}&lt;/math&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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;"><br /></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>: &lt;math&gt;\tilde P_1 = (x - x_2)(x - x_3) = (x - 1)(x - 3) = x^2 - 4x + 3&lt;/math&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>: &lt;math&gt;\tilde P_2 = (x - 0)(x - 3) = x^2 - 3x&lt;/math&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>: &lt;math&gt;\tilde P_3 = (x - 0)(x - 1) = x^2 - x&lt;/math&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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;"><br /></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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;"><br /></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>&lt;math&gt;\begin{matrix}</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>\tilde P_1(0) = 3 &amp; \tilde P_1(1) = 0 &amp; \tilde P_1(3) = 0 \\</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>\tilde P_2(0) = 0 &amp; \tilde P_2(1) = -2 &amp; \tilde P_2(3) = 0 \\</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>\tilde P_3(0) = 0 &amp; \tilde P_3(1) = 0 &amp; \tilde P_3(3) = 6 \\</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>\end{matrix}&lt;/math&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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;"><br /></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>Set &lt;math&gt;P_i(x) = \frac1{\tilde P_i(x)} \cdot \tilde P_i = \prod_{j \ne i} \frac{(x - x_j)}{(x_i - x_j)}&lt;/math&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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;"><br /></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>Then &lt;math&gt;P_i(x_k) = \begin{cases}</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>0 &amp; i \ne k \\</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>1 &amp; i = k \\</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>\end{cases} &lt;/math&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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;"><br /></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>: &lt;math&gt;P_1(x) = \frac13 x^2 - \frac43 x + 1&lt;/math&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>: &lt;math&gt;P_2(x) = -\frac12 x^2 + \frac32 x&lt;/math&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>: &lt;math&gt;P_3(x) = \frac16 x^2 - \frac16 x&lt;/math&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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;"><br /></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>Let &lt;math&gt;P = \sum_{i=1}^{n+1} y_i P_i(x) \in \mathbb P_n(\real)&lt;/math&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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;"><br /></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>: &lt;math&gt;P = 5 \cdot \left( \frac13 x^2 - \frac43 x + 1 \right) + 2 \cdot \left( -\frac12 x^2 + \frac32 x \right) + 2 \cdot \left( \frac16 x^2 - \frac16 x \right)&lt;/math&gt;</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>: &lt;math&gt;\,\! = x^2 - 4x + 5&lt;/math&gt;</div></td> </tr> </table>

C8sd https://drorbn.net/index.php?title=09-240/Classnotes_for_Tuesday_October_13&diff=8447&oldid=prev 2009-10-30T10:56:29Z

<p></p> <table style="background-color: #fff; color: #202122;" 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: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 06:56, 30 October 2009</td> </tr><tr> <td colspan="2" class="diff-lineno">Line 1:</td> <td colspan="2" class="diff-lineno">Line 1:</td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></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>{{09-240/Navigation}}</div></td> </tr> <tr> <td class="diff-marker"><a class="mw-diff-movedpara-left" title="Paragraph was moved. Click to jump to new location." href="#movedpara_3_0_rhs">&#x26AB;</a></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><a name="movedpara_1_0_lhs"></a>[[Image:Oct 13 notes page <del style="font-weight: bold; text-decoration: none;">1</del>.JPG]]</div></td> <td colspan="2" class="diff-empty diff-side-added"></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>[[Image:Oct 13 notes page <del style="font-weight: bold; text-decoration: none;">2</del>.JPG]]</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>[[Image:Oct 13 notes page <ins style="font-weight: bold; text-decoration: none;">1</ins>.JPG<ins style="font-weight: bold; text-decoration: none;">|600px</ins>]]</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>[[Image:Oct 13 notes page <del style="font-weight: bold; text-decoration: none;">3</del>.JPG]]</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>[[Image:Oct 13 notes page <ins style="font-weight: bold; text-decoration: none;">2</ins>.JPG<ins style="font-weight: bold; text-decoration: none;">|600px</ins>]]</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>[[Image:Oct 13 notes page <del style="font-weight: bold; text-decoration: none;">4</del>.JPG]]</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>[[Image:Oct 13 notes page <ins style="font-weight: bold; text-decoration: none;">3</ins>.JPG<ins style="font-weight: bold; text-decoration: none;">|600px</ins>]]</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker"><a class="mw-diff-movedpara-right" title="Paragraph was moved. Click to jump to old location." href="#movedpara_1_0_lhs">&#x26AB;</a></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><a name="movedpara_3_0_rhs"></a>[[Image:Oct 13 notes page <ins style="font-weight: bold; text-decoration: none;">4</ins>.JPG<ins style="font-weight: bold; text-decoration: none;">|600px</ins>]]</div></td> </tr> </table>

Drorbn https://drorbn.net/index.php?title=09-240/Classnotes_for_Tuesday_October_13&diff=8439&oldid=prev 2009-10-30T02:15:33Z

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