09-240/Classnotes for Tuesday September 22 - Revision history

C8sd: /* Class notes for today */ Format.

2009-09-24T21:49:35Z

Class notes for today: Format.

← Older revision		Revision as of 17:49, 24 September 2009
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	# can be multiplied by a number (not another vector)		# can be multiplied by a number (not another vector)

	Let ''F'' be a field. A vector space '''V''' over the field ''F'' is a set '''V''' (of vectors) with a special element 0<sub>'''V'''</sub>, a binary operation + : '''V''' × '''V''' → '''V''', a binary operation • : ''F'' × '''V''' → '''V'''.		Let ''F'' be a field. A vector space ''V'' over the field ''F'' is a set ''V'' (of vectors) with a special element 0<sub>''V''</sub>, a binary operation + : ''V'' × ''V'' → ''V'', a binary operation • : ''F'' × ''V'' → ''V''.

	{\| style="border: solid 1px black"		{\| style="border: solid 1px black"

C8sd: /* Class notes for today */ Format.

2009-09-24T17:52:27Z

Class notes for today: Format.

← Older revision		Revision as of 13:52, 24 September 2009
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	# can be multiplied by a number (not another vector)		# can be multiplied by a number (not another vector)

	Let ~~<math>\mathcal~~ F~~</math>~~ be a field. A vector space ~~<math>\mathbf~~ V~~</math>~~ over the field ~~<math>\mathcal~~ F~~</math>~~ is a set ~~<math>\mathbf~~ V~~</math>~~ (of vectors) with a special element <~~math~~>~~0_V~~</~~math~~>, a binary operation ~~<math>~~+ : ~~\mathbf~~ V ~~\times \mathbf~~ V ~~\rightarrow \mathbf~~ V~~</math>~~, a binary operation ~~<math>\cdot~~ : ~~\mathcal~~ F ~~\times \mathbf~~ V ~~\rightarrow \mathbf~~ V~~</math>~~.		Let ''F'' be a field. A vector space '''V''' over the field ''F'' is a set '''V''' (of vectors) with a special element 0<sub>'''V'''</sub>, a binary operation + : '''V''' × '''V''' → '''V''', a binary operation • : ''F'' × '''V''' → '''V'''.

	{\| style="border: solid 1px black"		{\| style="border: solid 1px black"
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	\| Convention for today:		\| Convention for today:
	: <math>x, y, z \in \mathbf V</math>		: <math>x, y, z \in \mathbf V</math>
	: <math>a, b, c \in ~~\mathcal~~ F</math>		: <math>a, b, c \in F</math>
	\|}		\|}

C8sd: /* Class notes for today */ Yangijay's notes

2009-09-23T02:46:32Z

Class notes for today: Yangijay's notes

← Older revision		Revision as of 22:46, 22 September 2009
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	{{09-240/Class Notes Warning}}		{{09-240/Class Notes Warning}}
	==Class notes for today==		==Class notes for today==

			<gallery>
			Image:September 22 2009 lecture notes page 1.jpg\|[[User:Yangjiay\|Yangjiay]] - Page 1
			Image:September 22 2009 lecture notes page 2.jpg\|Page 2
			Image:September 22 2009 lecture notes page 3.jpg\|Page 3
			Image:September 22 2009 lecture notes page 4.jpg\|Page 4
			Image:September 22 2009 lecture notes page 5.jpg\|Page 5
			</gallery>

	Vectors:		Vectors:

Physics: /* Food for thought */

2009-09-23T02:44:42Z

Food for thought

← Older revision		Revision as of 22:44, 22 September 2009
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	# Unnecessary for a V.S.		# Unnecessary for a V.S.
			# This is useless, since it does not describe reality. For example, a mathematical theory with 46 dimensions can be perfect and mathematically elegant, but if the only solution to it is a universe in which life cannot form it is not reality, hence we have no use for it.
	# This is useless

C8sd: /* Examples */

2009-09-22T21:50:10Z

Examples

← Older revision		Revision as of 17:50, 22 September 2009
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	# <math>\mathrm M_{m \times n}(F)</math>		# <math>\mathrm M_{m \times n}(F)</math>
	#: ...		#: ...
			# <math>\mathcal F(S, F)</math>
			# Polynomials
			# <math>...</math>

	=== Food for thought ===		=== Food for thought ===

C8sd: /* Proof of VS4 */ Incomplete examples, and food for thought

2009-09-22T21:46:22Z

Proof of VS4: Incomplete examples, and food for thought

← Older revision		Revision as of 17:46, 22 September 2009
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	Set <math>y = \begin{pmatrix} -a_1 \\ -a_2 \\ \vdots \\ -a_n \end{pmatrix}</math> and note		Set <math>y = \begin{pmatrix} -a_1 \\ -a_2 \\ \vdots \\ -a_n \end{pmatrix}</math> and note
	: <math>x + y = \begin{pmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end{pmatrix} + \begin{pmatrix} -a_1 \\ -a_2 \\ \vdots \\ -a_n \end{pmatrix} = \begin{pmatrix} a_1 + (-a_1) \\ a_2 + (-a_2) \\ \vdots \\ a_n + (-a_n) \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ \vdots \\ 0 \end{pmatrix} = 0_{F^n}</math>		: <math>x + y = \begin{pmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end{pmatrix} + \begin{pmatrix} -a_1 \\ -a_2 \\ \vdots \\ -a_n \end{pmatrix} = \begin{pmatrix} a_1 + (-a_1) \\ a_2 + (-a_2) \\ \vdots \\ a_n + (-a_n) \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ \vdots \\ 0 \end{pmatrix} = 0_{F^n}</math>

			=== Examples ===

			# <math>F^n \mbox{ for } n \in \mathbb N</math>
			#: <math>F^n = \left\{ \begin{pmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end{pmatrix} : a_i \in F \right\}</math>
			#: <math>\begin{pmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end{pmatrix} + \begin{pmatrix} b_1 \\ b_2 \\ \vdots \\ b_n \end{pmatrix} = \begin{pmatrix} a_1 + b_1 \\ a_2 + b_2 \\ \vdots \\ a_n + b_n \end{pmatrix}</math>
			#: <math>a \begin{pmatrix} b_1 \\ b_2 \\ \vdots \\ b_n \end{pmatrix} = \begin{pmatrix} ab_1 \\ ab_2 \\ \vdots \\ ab_n \end{pmatrix}</math>
			#: ...
			# <math>\mathrm M_{m \times n}(F)</math>
			#: ...

			=== Food for thought ===

			What is wrong with setting

			<math>
			\begin{pmatrix}
			2 & 3 \\
			4 & 5 \\
			\end{pmatrix} \cdot \begin{pmatrix}
			6 & 7 \\
			8 & 9 \\
			\end{pmatrix} = \begin{pmatrix}
			2 \cdot 6 & 3 \cdot 7 \\
			4 \cdot 8 & 5 \cdot 9 \\
			\end{pmatrix} = \begin{pmatrix}
			12 & 21 \\
			32 & 45 \\
			\end{pmatrix} ?
			</math>

			# Unnecessary for a V.S.
			# This is useless

C8sd: /* Class notes for today */ Add vector section before examples.

2009-09-22T21:24:52Z

Class notes for today: Add vector section before examples.

← Older revision		Revision as of 17:24, 22 September 2009
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	{{09-240/Class Notes Warning}}		{{09-240/Class Notes Warning}}
	==Class notes for today==		==Class notes for today==

			Vectors:
			# can be added
			# can be multiplied by a number (not another vector)

			Let <math>\mathcal F</math> be a field. A vector space <math>\mathbf V</math> over the field <math>\mathcal F</math> is a set <math>\mathbf V</math> (of vectors) with a special element <math>0_V</math>, a binary operation <math>+ : \mathbf V \times \mathbf V \rightarrow \mathbf V</math>, a binary operation <math>\cdot : \mathcal F \times \mathbf V \rightarrow \mathbf V</math>.

			{\| style="border: solid 1px black"
			\|-
			\| Convention for today:
			: <math>x, y, z \in \mathbf V</math>
			: <math>a, b, c \in \mathcal F</math>
			\|}

			VS1 <math>\forall x, y \in \mathbf V, x + y = y + x</math><br />
			VS2 <math>\cdots (x + y) + z = x + (y + z)</math><br />
			VS3 <math>\cdots x + 0 = x</math><br />
			VS4 <math>\forall x, \exists y \mbox{ s.t. } x + y = 0</math><br />
			VS5 <math>1 \cdot x = x</math><br />
			VS6 <math>a \cdot (b \cdot x) = (a \cdot b) \cdot x</math><br />
			VS7 <math>a \cdot (x + y) = ax + ay</math><br />
			VS8 <math>(a + b) \cdot x = ax + bx</math>

			=== Proof of VS4 ===

			Take an arbitrary <math>x = \begin{pmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end{pmatrix} \in F^n</math>

			Set <math>y = \begin{pmatrix} -a_1 \\ -a_2 \\ \vdots \\ -a_n \end{pmatrix}</math> and note
			: <math>x + y = \begin{pmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end{pmatrix} + \begin{pmatrix} -a_1 \\ -a_2 \\ \vdots \\ -a_n \end{pmatrix} = \begin{pmatrix} a_1 + (-a_1) \\ a_2 + (-a_2) \\ \vdots \\ a_n + (-a_n) \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ \vdots \\ 0 \end{pmatrix} = 0_{F^n}</math>

Drorbn at 00:38, 22 September 2009

2009-09-22T00:38:45Z

← Older revision		Revision as of 20:38, 21 September 2009
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	==Some links==		==Some links==
			* {{Pensieve Link\|2009-09/nb/09-240-TheComplexField.pdf\|The Complex Numbers by Computer}}.
	* Dori Eldar's work on "mechanical computations": {{Home Link\|People/Eldar/thesis/linkfunc.htm\|Machines as Calculating Devices}} and {{Home Link\|People/Eldar/thesis/squaring.htm\|Computing the function <math>W=Z^2</math> the hard way}}.		* Dori Eldar's work on "mechanical computations": {{Home Link\|People/Eldar/thesis/linkfunc.htm\|Machines as Calculating Devices}} and {{Home Link\|People/Eldar/thesis/squaring.htm\|Computing the function <math>W=Z^2</math> the hard way}}.
	* The "Dimensions" video on "Nombres complexes", is at http://dimensions-math.org/Dim_reg_AM.htm (and then go to "Dimensions_5".		* The "Dimensions" video on "Nombres complexes", is at http://dimensions-math.org/Dim_reg_AM.htm (and then go to "Dimensions_5".

Drorbn at 23:49, 21 September 2009

2009-09-21T23:49:27Z

← Older revision		Revision as of 19:49, 21 September 2009
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	* Dori Eldar's work on "mechanical computations": {{Home Link\|People/Eldar/thesis/linkfunc.htm\|Machines as Calculating Devices}} and {{Home Link\|People/Eldar/thesis/squaring.htm\|Computing the function <math>W=Z^2</math> the hard way}}.		* Dori Eldar's work on "mechanical computations": {{Home Link\|People/Eldar/thesis/linkfunc.htm\|Machines as Calculating Devices}} and {{Home Link\|People/Eldar/thesis/squaring.htm\|Computing the function <math>W=Z^2</math> the hard way}}.
	* The "Dimensions" video on "Nombres complexes", is at http://dimensions-math.org/Dim_reg_AM.htm (and then go to "Dimensions_5".		* The "Dimensions" video on "Nombres complexes", is at http://dimensions-math.org/Dim_reg_AM.htm (and then go to "Dimensions_5".


	{{09-240/Class Notes Warning}}		{{09-240/Class Notes Warning}}

Drorbn at 23:49, 21 September 2009

2009-09-21T23:49:16Z

New page

{{09-240/Navigation}}

==Some links==
* Dori Eldar's work on "mechanical computations": {{Home Link|People/Eldar/thesis/linkfunc.htm|Machines as Calculating Devices}} and {{Home Link|People/Eldar/thesis/squaring.htm|Computing the function <math>W=Z^2</math> the hard way}}.
* The "Dimensions" video on "Nombres complexes", is at http://dimensions-math.org/Dim_reg_AM.htm (and then go to "Dimensions_5".

{{09-240/Class Notes Warning}}
==Class notes for today==