14-240/Classnotes for Monday September 22 - Revision history

Boyang.wu: /* Scanned Lecture Notes by AM */

2014-12-08T05:57:26Z

Scanned Lecture Notes by AM

← Older revision		Revision as of 01:57, 8 December 2014
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	File:MAT240 Sept 22,14 (2 of 2).pdf\|page 2		File:MAT240 Sept 22,14 (2 of 2).pdf\|page 2
	</gallery>		</gallery>
			==Scanned Lecture Notes by [[User Boyang.wu\|Boyang.wu]]==

	[[File:W31.pdf]]		[[File:W31.pdf]]

Boyang.wu: /* Scanned Lecture Notes by AM */

2014-12-08T05:38:15Z

Scanned Lecture Notes by AM

← Older revision		Revision as of 01:38, 8 December 2014
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	File:MAT240 Sept 22,14 (2 of 2).pdf\|page 2		File:MAT240 Sept 22,14 (2 of 2).pdf\|page 2
	</gallery>		</gallery>
			[[File:W31.pdf]]

Hameeral: Format scanned lecture notes into a gallery.

2014-09-29T01:02:08Z

Format scanned lecture notes into a gallery.

← Older revision		Revision as of 21:02, 28 September 2014
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	* <math>VS_8 : \forall a, b \in F, \forall x \in V, (a + b)x = ax + bx</math>.		* <math>VS_8 : \forall a, b \in F, \forall x \in V, (a + b)x = ax + bx</math>.

			==Scanned Lecture Notes by [[User:AM\|AM]]==

			<gallery>
	http://drorbn.net/images/f/f0/MAT240_Sept_22%2C14_%281_of_2%29.pdf (Lecture Notes by AM 1 of 2)
			File:MAT240 Sept 22,14 (1 of 2).pdf\|page 1

			File:MAT240 Sept 22,14 (2 of 2).pdf\|page 2
	http://drorbn.net/images/4/48/MAT240_Sept_22%2C14_%282_of_2%29.pdf (Lecture notes by AM 2 of 2)
			</gallery>

AM at 12:51, 25 September 2014

2014-09-25T12:51:28Z

← Older revision		Revision as of 08:51, 25 September 2014
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	http://drorbn.net/images/f/f0/MAT240_Sept_22%2C14_%281_of_2%29.pdf		http://drorbn.net/images/f/f0/MAT240_Sept_22%2C14_%281_of_2%29.pdf (Lecture Notes by AM 1 of 2)

			http://drorbn.net/images/4/48/MAT240_Sept_22%2C14_%282_of_2%29.pdf (Lecture notes by AM 2 of 2)

AM at 12:49, 25 September 2014

2014-09-25T12:49:53Z

← Older revision		Revision as of 08:49, 25 September 2014
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	* <math>VS_4 : \forall x \in V, \exists y \in V, x + y = 0</math>.		* <math>VS_4 : \forall x \in V, \exists y \in V, x + y = 0</math>.
	* <math>VS_5 : \forall x \in V, 1 \times x = x</math>.		* <math>VS_5 : \forall x \in V, 1 \times x = x</math>.

	http://drorbn.net/skins/common/images/icons/fileicon-pdf.png
	* <math>VS_6 : \forall a, b \in F, \forall x \in V, a(bx) = (ab)x</math>.		* <math>VS_6 : \forall a, b \in F, \forall x \in V, a(bx) = (ab)x</math>.
	* <math>VS_7 : \forall a \in F, \forall x, y \in V, a(x + y) = ax + ay</math>.		* <math>VS_7 : \forall a \in F, \forall x, y \in V, a(x + y) = ax + ay</math>.
	* <math>VS_8 : \forall a, b \in F, \forall x \in V, (a + b)x = ax + bx</math>.		* <math>VS_8 : \forall a, b \in F, \forall x \in V, (a + b)x = ax + bx</math>.


			http://drorbn.net/images/f/f0/MAT240_Sept_22%2C14_%281_of_2%29.pdf

AM at 12:49, 25 September 2014

2014-09-25T12:49:08Z

← Older revision		Revision as of 08:49, 25 September 2014
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	* <math>VS_4 : \forall x \in V, \exists y \in V, x + y = 0</math>.		* <math>VS_4 : \forall x \in V, \exists y \in V, x + y = 0</math>.
	* <math>VS_5 : \forall x \in V, 1 \times x = x</math>.		* <math>VS_5 : \forall x \in V, 1 \times x = x</math>.

			http://drorbn.net/skins/common/images/icons/fileicon-pdf.png
	* <math>VS_6 : \forall a, b \in F, \forall x \in V, a(bx) = (ab)x</math>.		* <math>VS_6 : \forall a, b \in F, \forall x \in V, a(bx) = (ab)x</math>.
	* <math>VS_7 : \forall a \in F, \forall x, y \in V, a(x + y) = ax + ay</math>.		* <math>VS_7 : \forall a \in F, \forall x, y \in V, a(x + y) = ax + ay</math>.

Donna Tjandra at 03:25, 25 September 2014

2014-09-25T03:25:36Z

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			{{14-240/Navigation}}
	Polar coordinates:		Polar coordinates:
	* <math>r \times e^{i\theta} = r \times cos\theta + i \times rsin\theta</math>		* <math>r \times e^{i\theta} = r \times cos\theta + i \times rsin\theta</math>

Yue.Jiang at 03:04, 25 September 2014

2014-09-25T03:04:38Z

← Older revision		Revision as of 23:04, 24 September 2014
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	The Fundamantal Theorem of Algebra:		The Fundamantal Theorem of Algebra:
	<math>a_n \times z^{n} + a_n-1 \times z^{n-1} + \dots + a_0</math>		<math>a_n \times z^{n} + a_n-1 \times z^{n-1} + \dots + a_0</math>
	where <math>a_i \in C and a_i != 0</math> has a soluion <math>z \in C</math>		where <math>a_i \in C </math>and<math> a_i != 0</math> has a soluion <math>z \in C</math>
	In particular, <math>z^{2} - 1 = 0</math> has a solution.		In particular, <math>z^{2} - 1 = 0</math> has a solution.

Yue.Jiang at 03:03, 25 September 2014

2014-09-25T03:03:53Z

Yue.Jiang at 03:01, 25 September 2014

2014-09-25T03:01:46Z

← Older revision		Revision as of 23:01, 24 September 2014
Line 1:		Line 1:
	Polar coordinates:		Polar coordinates:
	* <math>r \times e^{i\theta} = r \times cos\theta + i \times rsin\theta</math>		* <math>r \times e^{i\theta} = r \times cos\theta + i \times rsin\theta</math>
	* <math>\r_1 \times e^{i\\theta_2} = r_1 \times (cos\theta + sin\theta</math>		* <math>r_1 \times e^{i\theta_2} = r_1 \times (cos\theta + sin\theta</math>

	The Fundamantal Theorem of Algebra:		The Fundamantal Theorem of Algebra:

← Older revision		Revision as of 23:03, 24 September 2014
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	The Fundamantal Theorem of Algebra:		The Fundamantal Theorem of Algebra:
	<math>\a_n \times z^{n} + \a_n-1 \times z^{n-1} + \dots + \a_0</math>		<math>a_n \times z^{n} + a_n-1 \times z^{n-1} + \dots + a_0</math>
	where <math>\a_i \in C and \a_i != 0</math> has a soluion <math>z \in C</math>		where <math>a_i \in C and a_i != 0</math> has a soluion <math>z \in C</math>
	In particular, <math>z^{2} - 1 = 0</math> has a solution.		In particular, <math>z^{2} - 1 = 0</math> has a solution.

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	Definition of Vector Space:		Definition of Vector Space:
	A "Vector Space" over a field F is a set V with a special element <math>\O_v \in V</math> and two binary operations:		A "Vector Space" over a field F is a set V with a special element <math>O_v \in V</math> and two binary operations:
	* <math>+ : V \times V -> V</math>		* <math>+ : V \times V -> V</math>
	* <math>\times : V \times V -> V</math>		* <math>\times : V \times V -> V</math>

	s.t.		s.t.
	* <math>\VS_1 : \forall x, y \in V, x + y = y + x</math>.		* <math>VS_1 : \forall x, y \in V, x + y = y + x</math>.
	* <math>\VS_2 : \forall x, y, z \in V, x + (y + z) = (x + y) + z</math>.		* <math>VS_2 : \forall x, y, z \in V, x + (y + z) = (x + y) + z</math>.
	* <math>\VS_3 : \forall x \in V, x + 0 = x</math>.		* <math>VS_3 : \forall x \in V, x + 0 = x</math>.
	* <math>\VS_4 : \forall x \in V, \exists y \in V, x + y = 0</math>.		* <math>VS_4 : \forall x \in V, \exists y \in V, x + y = 0</math>.
	* <math>\VS_5 : \forall x \in V, 1 \times x = x</math>.		* <math>VS_5 : \forall x \in V, 1 \times x = x</math>.
	* <math>\VS_6 : \forall a, b \in F, \forall x \in V, a(bx) = (ab)x</math>.		* <math>VS_6 : \forall a, b \in F, \forall x \in V, a(bx) = (ab)x</math>.
	* <math>\VS_7 : \forall a \in F, \forall x, y \in V, a(x + y) = ax + ay</math>.		* <math>VS_7 : \forall a \in F, \forall x, y \in V, a(x + y) = ax + ay</math>.
	* <math>\VS_8 : \forall a, b \in F, \forall x \in V, (a + b)x = ax + bx</math>.		* <math>VS_8 : \forall a, b \in F, \forall x \in V, (a + b)x = ax + bx</math>.