06-240/Classnotes For Thursday, September 28 - Revision history

Wongpak at 02:25, 25 October 2006

2006-10-25T02:25:27Z

← Older revision		Revision as of 22:25, 24 October 2006
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	** note2:[[http://i98.photobucket.com/albums/l269/uhoang/2.jpg]]		** note2:[[http://i98.photobucket.com/albums/l269/uhoang/2.jpg]]

	===Scan of Lecture notes===

	* PDF file by [[User:Alla]]: [[Media:MAT_Lect006.pdf\|Week 3 Lecture 2 notes]]		* PDF file by [[User:Alla]]: [[Media:MAT_Lect006.pdf\|Week 3 Lecture 2 notes]]
			* PDF file by [[User:Gokmen]]: [[Media:06-240-Lecture-28-september.pdf\|Week 3 Lecture 2 notes]]

	===Scan of Tutorial notes===		===Scan of Tutorial notes===

	* PDF file by [[User:Alla]]: [[Media:MAT_Tut003.pdf\|Week 3 Tutorial notes]]		* PDF file by [[User:Alla]]: [[Media:MAT_Tut003.pdf\|Week 3 Tutorial notes]]
			* PDF file by [[User:Gokmen]]: [[Media:06-240-tutorial-28-september.pdf\|Week 3 Tutorial notes]]

	===Linear Combination===		===Linear Combination===

Drorbn at 23:37, 2 October 2006

2006-10-02T23:37:50Z

← Older revision		Revision as of 19:37, 2 October 2006
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			{{06-240/Navigation}}

	===Scan of Lecture notes===		===Scan of Lecture notes===

	*Image file: week 3 lecture		*Image file: week 3 lecture
	-note1[[http://i98.photobucket.com/albums/l269/uhoang/1.jpg]]		** note1[[http://i98.photobucket.com/albums/l269/uhoang/1.jpg]]
	-note2:[[http://i98.photobucket.com/albums/l269/uhoang/2.jpg]]		** note2:[[http://i98.photobucket.com/albums/l269/uhoang/2.jpg]]

	===Scan of Lecture notes===		===Scan of Lecture notes===

Uhoang36: /* Scan of Lecture notes */

2006-10-02T01:06:53Z

Scan of Lecture notes

← Older revision		Revision as of 21:06, 1 October 2006
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			===Scan of Lecture notes===

			*Image file: week 3 lecture
			-note1[[http://i98.photobucket.com/albums/l269/uhoang/1.jpg]]
			-note2:[[http://i98.photobucket.com/albums/l269/uhoang/2.jpg]]

	===Scan of Lecture notes===		===Scan of Lecture notes===

Alla: Added scan of Week 3 Tutorial notes

2006-10-01T03:00:06Z

Added scan of Week 3 Tutorial notes

← Older revision		Revision as of 23:00, 30 September 2006
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	* PDF file by [[User:Alla]]: [[Media:MAT_Lect006.pdf\|Week 3 Lecture 2 notes]]		* PDF file by [[User:Alla]]: [[Media:MAT_Lect006.pdf\|Week 3 Lecture 2 notes]]

			===Scan of Tutorial notes===

			* PDF file by [[User:Alla]]: [[Media:MAT_Tut003.pdf\|Week 3 Tutorial notes]]

	===Linear Combination===		===Linear Combination===

Alla: Added scan of Week 3 Lecture 2 notes

2006-10-01T02:58:18Z

Added scan of Week 3 Lecture 2 notes

← Older revision		Revision as of 22:58, 30 September 2006
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			===Scan of Lecture notes===

			* PDF file by [[User:Alla]]: [[Media:MAT_Lect006.pdf\|Week 3 Lecture 2 notes]]

	===Linear Combination===		===Linear Combination===

Mc: grammar fixup

2006-09-30T15:41:51Z

grammar fixup

← Older revision		Revision as of 11:41, 30 September 2006
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	<math>\mbox{Solve the four equations above and we will get }a_1^{}=-4\mbox{ and }a_2^{}=2</math>		<math>\mbox{Solve the four equations above and we will get }a_1^{}=-4\mbox{ and }a_2^{}=2</math>

	<math>\mbox{Check if }a_1^{}=-4\mbox{ and }a_2^{}=2\mbox{ ~~hold~~ for all ~~the~~ 4 equations.}</math>		<math>\mbox{Check if }a_1^{}=-4\mbox{ and }a_2^{}=2\mbox{ holds for all 4 equations.}</math>

	<math>\mbox{Since it holds, } v\in W</math>		<math>\mbox{Since it holds, } v\in W</math>

Mc: LaTeX cleanup

2006-09-30T15:39:54Z

LaTeX cleanup

Show changes

Wongpak at 00:59, 30 September 2006

2006-09-30T00:59:26Z

@@ Line 33: / Line 33: @@
 ''Example''
-. Let P<sub>3</sub>(<math>\Re</math>)={ax<sup>3</sup>+bx<sup>2</sup>+cx+d}<math>\subseteq</math>P(<math>\Re</math>), ''a'', ''b'', ''c'', ''d'', <math>\in \Re</math>.
+. Let P<sub>3</sub>(<math>\Re</math>)={ax<sup>3</sup>+bx<sup>2</sup>+cx+d}<math>\subseteq</math>P(<math>\Re</math>), ''a'', ''b'', ''c'', ''d'', <math>\in \Re</math>.<BR>
-''u''<sub>1</sub>=''x''<sup>3</sup>-2''x''<sup>2</sup>-5''x''-3
+''u''<sub>1</sub>=''x''<sup>3</sup>-2''x''<sup>2</sup>-5''x''-3<BR>
-''u''<sub>2</sub>=3''x''<sup>3</sup>-5''x''<sup>2</sup>-4''x''-9
+''u''<sub>2</sub>=3''x''<sup>3</sup>-5''x''<sup>2</sup>-4''x''-9<BR>
-''v''=2''x''<sup>3</sup>-2''x''<sup>2</sup>+12''x''-6
+''v''=2''x''<sup>3</sup>-2''x''<sup>2</sup>+12''x''-6<BR>

74.117.92.192: /* Span */

2006-09-29T20:25:31Z

Span

← Older revision		Revision as of 16:25, 29 September 2006
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			''Example''
	To be continued ...
			1. Let P<sub>3</sub>(<math>\Re</math>)={ax<sup>3</sup>+bx<sup>2</sup>+cx+d}<math>\subseteq</math>P(<math>\Re</math>), ''a'', ''b'', ''c'', ''d'', <math>\in \Re</math>.
			''u''<sub>1</sub>=''x''<sup>3</sup>-2''x''<sup>2</sup>-5''x''-3
			''u''<sub>2</sub>=3''x''<sup>3</sup>-5''x''<sup>2</sup>-4''x''-9
			''v''=2''x''<sup>3</sup>-2''x''<sup>2</sup>+12''x''-6

Wongpak: /* Span */

2006-09-29T18:07:46Z

Span

← Older revision		Revision as of 14:07, 29 September 2006
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	<math>\Rightarrow</math> ''x''+''y'' = <math>\sum_{i=1}^n</math> ''a''<sub>i</sub>''u''<sub>i</sub> + <math>\sum_{i=1}^m</math> ''b''<sub>i</sub>''v''<sub>i</sub> = <math>\sum_{i=1}^{m+n}</math> ''c''<sub>i</sub>''w''<sub>i</sub> where ''c''<sub>i</sub>=(''a''<sub>1</sub>, ''a''<sub>2</sub>,...,''a''<sub>n</sub>, ''b''<sub>1</sub>, ''b''<sub>2</sub>,...,''b''<sub>m</sub>) and ''w''<sub>i</sub>=''c''<sub>i</sub>=(''u''<sub>1</sub>, ''u''<sub>2</sub>,...,''u''<sub>n</sub>, ''v''<sub>1</sub>, ''v''<sub>2</sub>,...,''v''<sub>m</sub>).<br>		<math>\Rightarrow</math> ''x''+''y'' = <math>\sum_{i=1}^n</math> ''a''<sub>i</sub>''u''<sub>i</sub> + <math>\sum_{i=1}^m</math> ''b''<sub>i</sub>''v''<sub>i</sub> = <math>\sum_{i=1}^{m+n}</math> ''c''<sub>i</sub>''w''<sub>i</sub> where ''c''<sub>i</sub>=(''a''<sub>1</sub>, ''a''<sub>2</sub>,...,''a''<sub>n</sub>, ''b''<sub>1</sub>, ''b''<sub>2</sub>,...,''b''<sub>m</sub>) and ''w''<sub>i</sub>=''c''<sub>i</sub>=(''u''<sub>1</sub>, ''u''<sub>2</sub>,...,''u''<sub>n</sub>, ''v''<sub>1</sub>, ''v''<sub>2</sub>,...,''v''<sub>m</sub>).<br>
	3. ''cx''= c<math>\sum_{i=1}^n</math> ''a''<sub>i</sub>''u''<sub>i</sub>=<math>\sum_{i=1}^n</math> (''ca''<sub>i</sub>)''u''<sub>i</sub><math>\in </math> span <math>\mathcal{S}</math>.		3. ''cx''= c<math>\sum_{i=1}^n</math> ''a''<sub>i</sub>''u''<sub>i</sub>=<math>\sum_{i=1}^n</math> (''ca''<sub>i</sub>)''u''<sub>i</sub><math>\in </math> span <math>\mathcal{S}</math>.


			To be continued ...