09-240/Classnotes for Tuesday December 1 - Revision history

C8sd: Format, correct mistakes.

2009-12-08T04:24:45Z

Format, correct mistakes.

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	MAT240 – December 1st		MAT240 – December 1st

	Basic Properties of ~~det: M~~<~~sub~~>~~nxn</sub>→F~~: 0 ~~det(I)~~ = 1		Basic Properties of <math>\det : \mathbb M_{n \times n} \rightarrow F</math>:

	1. ~~<math>~~det(E'~~_{i,j\,\!}A~~) = -det(A) ; ~~\|E'_{i,j\,\!}\|= -1. [Note:~~ det(EA) = \|~~E\|\|~~A\|~~]</math>~~		(Note that det(''EA'') = det(''E'')·det(''A'') and that det(''A'') may be written as \|''A''\|.)

			0. <math>\,\! \det(I) = 1</math>
⚫	* ~~Also, note that exchanging~~ two rows flips the sign.

	2. <math>det(E^2_{i,~~c\,\!~~}A) = det(A) ; \|E^2_{i,j~~,c\,\!~~}\| = 1</math>		1. <math>\det(E^1_{i,j}A) = -\det(A) ; \|E^1_{i,j}\|= -1</math>

		⚫	: Exchanging two rows flips the sign.
⚫	* These are "enough"!

	3. <math>det(~~(E_~~{i,j,c~~\,\!~~}A) = det(A) ; \|E^3_{i,j,c~~\,\!~~}\| = 1</math>		2. <math>\det(E^2_{i,c}A) = c \cdot \det(A) ; \|E^2_{i,c}\| = c</math>

		⚫	: These are "enough"!
⚫	* Adding a multiple of one row to another does not change the determinant.

			3. <math>\det(E^3_{i,j,c}A) = \det(A) ; \|E^3_{i,j,c}\| = 1</math>

		⚫	: Adding a multiple of one row to another does not change the determinant.

	The determinant of any matrix can be calculated using the properties above.		The determinant of any matrix can be calculated using the properties above.

	~~<b>~~Theorem:~~</b>~~		'''Theorem''':

	If <math> det' : M_{~~nxn~~\~~,\!~~}</math>→F satisfies properties 0-3 above, then <math>det' = det</math>		If <math>{\det}' : \mathbb M_{n \times n} \rightarrow F</math> satisfies properties 0-3 above, then <math>\det' = \det</math>

	<math>det(A) = det'(A)</math>		<math>\det(A) = \det'(A)</math>

	Philosophical remark: Why not begin our inquiry with the properties above?		Philosophical remark: Why not begin our inquiry with the properties above?

	We must find an implied need for their use; thus, we must know whether a function <math>det</math> exists first.		We must find an implied need for their use; thus, we must know whether a function <math>\det</math> exists first.

Kjgry at 04:06, 5 December 2009

2009-12-05T04:06:22Z

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	</gallery>		</gallery>

	In the above gallery, there is a complete copy of notes for the lecture given on December 1st by Professor Natan (in PDF format).		~In the above gallery, there is a complete copy of notes for the lecture given on December 1st by Professor Natan (in PDF format).

	--- Wiki Format ---		--- Wiki Format ---

Kjgry: Uploaded PDF for Dec 1st and created a gallery.

2009-12-05T04:05:38Z

Uploaded PDF for Dec 1st and created a gallery.

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	{{09-240/Navigation}}		{{09-240/Navigation}}
			<gallery>
			Image:ALA240-2009_-_December_1st.pdf\|A complete set of the December 1st lecture notes given by Professor Dror Bar-Natan for the Fall Session of MAT240 at the University of Toronto.
		⚫	Image:Dec 1 lecture notes Pg 1.JPG\|
		⚫	Image:Dec 1 lecture notes Pg 2.JPG\|
		⚫	Image:Dec 1 lecture notes Pg 3.JPG\|
		⚫	Image:Dec 1 lecture notes Pg 4.JPG\|
		⚫	Image:Dec 1 lecture notes Pg 5.JPG\|
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		⚫	Image:dec1-7.jpg\|
		⚫	Image:dec1-8.jpg\|
			</gallery>

			In the above gallery, there is a complete copy of notes for the lecture given on December 1st by Professor Natan (in PDF format).
⚫	[[Image:Dec 1 lecture notes Pg 1.JPG\|~~600px]]~~
⚫	[[Image:Dec 1 lecture notes Pg 2.JPG\|~~600px]]~~
⚫	[[Image:Dec 1 lecture notes Pg 3.JPG\|~~600px]]~~
⚫	[[Image:Dec 1 lecture notes Pg 4.JPG\|~~600px]]~~
⚫	[[Image:Dec 1 lecture notes Pg 5.JPG\|~~600px]]~~
⚫	[[Image:dec1-1.jpg\|~~700px]]~~
⚫	[[Image:dec1-2.jpg\|~~700px]]~~
⚫	[[Image:dec1-3.jpg\|~~700px]]~~
⚫	[[Image:dec1-4.jpg\|~~700px]]~~
⚫	[[Image:dec1-5.jpg\|~~700px]]~~
⚫	[[Image:dec1-6.jpg\|~~700px]]~~
⚫	[[Image:dec1-7.jpg\|~~700px]]~~
⚫	[[Image:dec1-8.jpg\|~~700px]]~~

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Milcak at 01:46, 4 December 2009

2009-12-04T01:46:29Z

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			{{09-240/Navigation}}

	[[Image:Dec 1 lecture notes Pg 1.JPG\|600px]]		[[Image:Dec 1 lecture notes Pg 1.JPG\|600px]]
	[[Image:Dec 1 lecture notes Pg 2.JPG\|600px]]		[[Image:Dec 1 lecture notes Pg 2.JPG\|600px]]

Mantynen at 21:31, 1 December 2009

2009-12-01T21:31:59Z

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	* Also, note that exchanging two rows flips the sign.		* Also, note that exchanging two rows flips the sign.

	2. <math>det(E^2_{i,c\,\!}A) = det(A) ; \|(E^2_{i,j,c\,\!}\| = 1</math>		2. <math>det(E^2_{i,c\,\!}A) = det(A) ; \|E^2_{i,j,c\,\!}\| = 1</math>

	* These are "enough"!		* These are "enough"!

	3. <math>det((E_{i,j,c\,\!}A) = det(A) ; \|(E^3_{i,j,c\,\!}\| = 1</math>		3. <math>det((E_{i,j,c\,\!}A) = det(A) ; \|E^3_{i,j,c\,\!}\| = 1</math>

	* Adding a multiple of one row to another does not change the determinant.		* Adding a multiple of one row to another does not change the determinant.
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	<b>Theorem:</b>		<b>Theorem:</b>

			If <math> det' : M_{nxn\,\!}</math>→F satisfies properties 0-3 above, then <math>det' = det</math>

			<math>det(A) = det'(A)</math>

			Philosophical remark: Why not begin our inquiry with the properties above?

			We must find an implied need for their use; thus, we must know whether a function <math>det</math> exists first.

Mantynen at 21:20, 1 December 2009

2009-12-01T21:20:29Z

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	* These are "enough"!		* These are "enough"!

	3. <math>det((E_{i,j,c\,\!}A) = det(A) ; \|(E^3_{i,j,c\,\!}\| = 1		3. <math>det((E_{i,j,c\,\!}A) = det(A) ; \|(E^3_{i,j,c\,\!}\| = 1</math>

	* Adding a multiple of one row to another does not change the determinant.		* Adding a multiple of one row to another does not change the determinant.

	The determinant of any matrix can be calculated using the properties above.~~</math>~~		The determinant of any matrix can be calculated using the properties above.

			<b>Theorem:</b>

Mantynen at 21:19, 1 December 2009

2009-12-01T21:19:37Z

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	MAT240 – December 1st		MAT240 – December 1st

	Basic Properties of det: M<sub>nxn</sub>		Basic Properties of det: M<sub>nxn</sub>→F: 0 det(I) = 1

			1. <math>det(E'_{i,j\,\!}A) = -det(A) ; \|E'_{i,j\,\!}\|= -1. [Note: det(EA) = \|E\|\|A\|]</math>

			* Also, note that exchanging two rows flips the sign.

			2. <math>det(E^2_{i,c\,\!}A) = det(A) ; \|(E^2_{i,j,c\,\!}\| = 1</math>

			* These are "enough"!

			3. <math>det((E_{i,j,c\,\!}A) = det(A) ; \|(E^3_{i,j,c\,\!}\| = 1

			* Adding a multiple of one row to another does not change the determinant.

			The determinant of any matrix can be calculated using the properties above.</math>

Mantynen at 20:54, 1 December 2009

2009-12-01T20:54:39Z

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			--- Wiki Format ---

			MAT240 – December 1st

			Basic Properties of det: M<sub>nxn</sub>

Mantynen at 20:44, 1 December 2009

2009-12-01T20:44:49Z

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Bright at 20:19, 1 December 2009

2009-12-01T20:19:25Z

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