06-240/Classnotes For Tuesday October 3 - Revision history

Drorbn at 12:14, 12 July 2007

2007-07-12T12:14:28Z

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	'''Example.''' In <math>P_3(\mathbb{R})</math>, <math>v_1=2x^3-2x^2+12-6</math> is a linear combination of: <math>u_1=x^3-2x^2-5x-3</math> and <math>u_2=3x^3-5x^2-4x-9</math> but <math>v_2=3x^3-2x^2+7x+8</math> is not.		'''Example.''' In <math>P_3(\mathbb{R})</math>, <math>v_1=2x^3-2x^2+12-6</math> is a linear combination of: <math>u_1=x^3-2x^2-5x-3</math> and <math>u_2=3x^3-5x^2-4x-9</math> but <math>v_2=3x^3-2x^2+7x+8</math> is not.

	<math>~~\mbox{Why?}{}_{}~~^{}</math>		'''Why?''' <math>v_1=2x^3-2x^2+12-6=a_1u_1+a_2u_2</math>
		⚫	<math>=a_1(x^3-2x^2-5x-3)+a_2(3x^3-5x^2-4x-9)</math>

	<math>v_1^{}=2x^3-2x^2+12-6=a_1^{}u_1+a_2u_2</math>

⚫	<math>=a_1(x^3-2x^2-5x-3)+a_2(3x^3-5x^2-4x-9)~~{}_{}^{}~~</math>

	<math>v_1^{}=-4u_1+2u_2</math>		<math>v_1^{}=-4u_1+2u_2</math>

Drorbn: /* Links to Classnotes */

2007-07-12T11:15:23Z

Links to Classnotes

← Older revision		Revision as of 07:15, 12 July 2007
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	'''Definition.''' <math>v\in V</math>is a linear combination of elements in <math>S\subset V</math> if <math>\exists u_1,\ldots,u_n\in S</math> and <math>a_1,\dots,a_n \in F</math> such that <math>V=\sum a_i u_i</math>		'''Definition.''' <math>v\in V</math>is a linear combination of elements in <math>S\subset V</math> if <math>\exists u_1,\ldots,u_n\in S</math> and <math>a_1,\dots,a_n \in F</math> such that <math>V=\sum a_i u_i</math>

			'''Example.''' In <math>P_3(\mathbb{R})</math>, <math>v_1=2x^3-2x^2+12-6</math> is a linear combination of: <math>u_1=x^3-2x^2-5x-3</math> and <math>u_2=3x^3-5x^2-4x-9</math> but <math>v_2=3x^3-2x^2+7x+8</math> is not.
	'''Example.''' <math>\mbox{In }P_3(\mathbb{R})\mbox{,}</math>
	<math>v_1^{}=2x^3-2x^2+12-6 \mbox{ is a linear combination of:}</math>
	<math>u_1^{}=x^3-2x^2-5x-3\mbox{ and }u_2=3x^3-5x^2-4x-9</math>
	<math>\mbox{but } v_2^{}=3x^3-2x^2+7x+8 \mbox{ is not.}</math>

	<math>\mbox{Why?}{}_{}^{}</math>		<math>\mbox{Why?}{}_{}^{}</math>

Drorbn: /* Links to Classnotes */

2007-07-12T02:35:56Z

Links to Classnotes

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	----		----
	'''Definition.''' <math>v\in V</math>is a linear combination of elements in <math>S\subset V</math>		'''Definition.''' <math>v\in V</math>is a linear combination of elements in <math>S\subset V</math> if <math>\exists u_1,\ldots,u_n\in S</math> and <math>a_1,\dots,a_n \in F</math> such that <math>V=\sum a_i u_i</math>
	<math> \mbox{ if }\exists u_1,\ldots,u_n\in S \mbox{ and } a_1,\dots,a_n \in F \mbox{ such that } V=\sum a_i u_i</math>

	<math>\mbox{~~Example~~}{}~~_{}^~~{}</math>		'''Example.''' <math>\mbox{In }P_3(\mathbb{R})\mbox{,}</math>

	<math>\mbox{In }P_3(\mathbb{R})\mbox{,}</math>
	<math>v_1^{}=2x^3-2x^2+12-6 \mbox{ is a linear combination of:}</math>		<math>v_1^{}=2x^3-2x^2+12-6 \mbox{ is a linear combination of:}</math>
	<math>u_1^{}=x^3-2x^2-5x-3\mbox{ and }u_2=3x^3-5x^2-4x-9</math>		<math>u_1^{}=x^3-2x^2-5x-3\mbox{ and }u_2=3x^3-5x^2-4x-9</math>

Drorbn at 02:33, 12 July 2007

2007-07-12T02:33:12Z

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	----		----
		⚫	'''Definition.''' <math>v\in V</math>is a linear combination of elements in <math>S\subset V</math>
	<math>\mbox{Definition}{}_{}^{}</math>

⚫	<math>v\in V ~~\mbox{~~ is a linear combination of elements in } S\subset V</math>
	<math> \mbox{ if }\exists u_1,\ldots,u_n\in S \mbox{ and } a_1,\dots,a_n \in F \mbox{ such that } V=\sum a_i u_i</math>		<math> \mbox{ if }\exists u_1,\ldots,u_n\in S \mbox{ and } a_1,\dots,a_n \in F \mbox{ such that } V=\sum a_i u_i</math>

Wongpak at 02:23, 25 October 2006

2006-10-25T02:23:36Z

← Older revision		Revision as of 22:23, 24 October 2006
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	<math>\mbox{Why?}{}_{}^{}</math>		<math>\mbox{Why?}{}_{}^{}</math>

	<math>v_1^{}=2x^3-2x^2+12-6=a_1^{}u_1+a_2u_2~~=a_1(x^3-2x^2-5x-3)+a_2(3x^3-5x^2-4x-9)~~</math>		<math>v_1^{}=2x^3-2x^2+12-6=a_1^{}u_1+a_2u_2</math>

			<math>=a_1(x^3-2x^2-5x-3)+a_2(3x^3-5x^2-4x-9){}_{}^{}</math>

	<math>v_1^{}=-4u_1+2u_2</math>		<math>v_1^{}=-4u_1+2u_2</math>

	<math>\mbox{Definition}{}_{}^{}</math>		<math>\mbox{Definition}{}_{}^{}</math>

	<math>\mbox{We say that a subset }S\subset V\mbox{ generates or spans }V ~~\mbox{ if span }S=\lbrace\mbox{ all linear combinations of elements in } S\rbrace=V~~</math>		<math>\mbox{We say that a subset }S\subset V\mbox{ generates or spans }V </math>

			<math>\mbox{ if span }S=\lbrace\mbox{ all linear combinations of elements in } S\rbrace=V{}_{}^{}</math>

	<math>\mbox{Examples}{}_{}^{}</math>		<math>\mbox{Examples}{}_{}^{}</math>

Wongpak: /* Links to Classnotes */

2006-10-25T02:10:48Z

Links to Classnotes

← Older revision		Revision as of 22:10, 24 October 2006
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	===Links to Classnotes===		===Links to Classnotes===
	* PDF file by [[User:Alla]]: [[Media:MAT_Lect007.pdf\|Week 4 Lecture 1 notes]]		* PDF file by [[User:Alla]]: [[Media:MAT_Lect007.pdf\|Week 4 Lecture 1 notes]]
			* PDF file by [[User:Gokmen]]: [[Media:06-240-Lecture-03-October.pdf\|Week 4 Lecture 1 notes]]

	----		----
	<math>\mbox{Definition}{}_{}^{}</math>		<math>\mbox{Definition}{}_{}^{}</math>

Wongpak at 02:09, 25 October 2006

2006-10-25T02:09:59Z

← Older revision		Revision as of 22:09, 24 October 2006
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			{{06-240/Navigation}}

	===Links to Classnotes===		===Links to Classnotes===
	* PDF file by [[User:Alla]]: [[Media:MAT_Lect007.pdf\|Week 4 Lecture 1 notes]]		* PDF file by [[User:Alla]]: [[Media:MAT_Lect007.pdf\|Week 4 Lecture 1 notes]]

Alla: Added scan of Week 4 Lecture 1 notes

2006-10-08T21:49:44Z

Added scan of Week 4 Lecture 1 notes

← Older revision		Revision as of 17:49, 8 October 2006
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			===Links to Classnotes===
			* PDF file by [[User:Alla]]: [[Media:MAT_Lect007.pdf\|Week 4 Lecture 1 notes]]
			----
	<math>\mbox{Definition}{}_{}^{}</math>		<math>\mbox{Definition}{}_{}^{}</math>

Mc: Lecture Notes (MC)

2006-10-07T22:34:02Z

Lecture Notes (MC)

New page

<math>\mbox{Definition}{}_{}^{}</math>

<math>v\in V \mbox{ is a linear combination of elements in } S\subset V</math>
<math> \mbox{ if }\exists u_1,\ldots,u_n\in S \mbox{ and } a_1,\dots,a_n \in F \mbox{ such that } V=\sum a_i u_i</math>

<math>\mbox{Example}{}_{}^{}</math>

<math>\mbox{In }P_3(\mathbb{R})\mbox{,}</math>
<math>v_1^{}=2x^3-2x^2+12-6 \mbox{ is a linear combination of:}</math>
<math>u_1^{}=x^3-2x^2-5x-3\mbox{ and }u_2=3x^3-5x^2-4x-9</math>
<math>\mbox{but } v_2^{}=3x^3-2x^2+7x+8 \mbox{ is not.}</math>

<math>\mbox{Why?}{}_{}^{}</math>

<math>v_1^{}=2x^3-2x^2+12-6=a_1^{}u_1+a_2u_2=a_1(x^3-2x^2-5x-3)+a_2(3x^3-5x^2-4x-9)</math>
<math>v_1^{}=-4u_1+2u_2</math>

<math>\mbox{Definition}{}_{}^{}</math>

<math>\mbox{We say that a subset }S\subset V\mbox{ generates or spans }V \mbox{ if span }S=\lbrace\mbox{ all linear combinations of elements in } S\rbrace=V</math>

<math>\mbox{Examples}{}_{}^{}</math>

<math>V=M_{2\times 2}(\mathbb{R})</math>

<math>M_1=\begin{pmatrix}1&0\\0&0\end{pmatrix},
M_2=\begin{pmatrix}0&1\\0&0\end{pmatrix},
M_3=\begin{pmatrix}0&0\\1&0\end{pmatrix},
M_4\begin{pmatrix}0&0\\0&1\end{pmatrix}</math>

<math>N_1=\begin{pmatrix}0&1\\1&1\end{pmatrix},
N_2=\begin{pmatrix}1&0\\1&1\end{pmatrix},
N_3=\begin{pmatrix}1&1\\0&1\end{pmatrix},
N_4\begin{pmatrix}1&1\\1&0\end{pmatrix}</math>

<math>\mbox{Claims}{}_{}^{}</math>

#<math>\lbrace M_1^{},M_2,M_3,M_4\rbrace\mbox{ generates }V</math>
#<math>\lbrace N_1^{},N_2,N_3,N_4\rbrace\mbox{ generates }V</math>
#<math>\lbrace M_1^{},M_2,M_3\rbrace\mbox{ does not generate }V</math>
#<math>\lbrace N_1^{},N_2,N_3\rbrace\mbox{ does not generate }V</math>
 
<math>\mbox{Proof of 1}{}_{}^{}</math>

<math>\mbox{Given any }B=\begin{pmatrix}b_{11}^{}&b_{12}\\b_{21}&b_{22}\end{pmatrix}\mbox{ need to find }a_1,a_2,a_3,a_4\mbox{ such that,}</math>

<math>\begin{pmatrix}b_{11}^{}&b_{12}\\b_{21}&b_{22}\end{pmatrix}=B=a_1M_1+a_2M_2+a_3M_3+a_4M_4=\begin{pmatrix}a_1&0\\0&0\end{pmatrix}
+\begin{pmatrix}0&a_2\\0&0\end{pmatrix}
+\begin{pmatrix}0&0\\a_3&0\end{pmatrix}
+\begin{pmatrix}0&0\\0&a_4\end{pmatrix}</math>

<math>=\begin{pmatrix}a_1^{}&a_2\\a_3&a_4\end{pmatrix}\Leftrightarrow
\begin{cases}b_{11}=a_1\\b_{12}=a_2\\b_{21}=a_3\\b_{22}=a_4\end{cases}</math>
<math>\mbox{A system of 4 equations with 4 unknowns}{}_{}^{}</math>
 
<math>\mbox{Proof of 2}{}_{}^{}</math>

<math>\begin{pmatrix}b_{11}^{}&b_{12}\\b_{21}&b_{22}\end{pmatrix}
=B=a_1N_1+a_2N_2+a_3N_3+a_4N_4=
\begin{pmatrix}0&a_1\\a_1&a_1\end{pmatrix}
+\begin{pmatrix}a_2&0\\a_2&a_2\end{pmatrix}
+\begin{pmatrix}a_3&a_3\\0&a_3\end{pmatrix}
+\begin{pmatrix}a_4&a_4\\a_4&0\end{pmatrix}</math>

<math>=\begin{pmatrix}a_2^{}+a_3+a_4&a_1+a_3+a_4\\a_1+a_2+a_4&a_1+a_2+a_3\end{pmatrix}\Leftrightarrow
\begin{cases}b_{11}=a_2+a_3+a_4\\b_{12}=a_1+a_3+a_4\\b_{21}=a_1+a_2+a_4\\b_{22}=a_1+a_2+a_3\end{cases}</math>

<math>\mbox{Trick}{}_{}^{}</math>

<math>M_1=\frac{1}{3}\left(N_1+N_2+N_3+N_4\right)-3N_1</math>
<math>M_2=\frac{1}{3}\left(N_1+N_2+N_3+N_4\right)-3N_2</math>
<math>M_3=\frac{1}{3}\left(N_1+N_2+N_3+N_4\right)-3N_3</math>
<math>M_4=\frac{1}{3}\left(N_1+N_2+N_3+N_4\right)-3N_4</math>

<math>B=b_{11}^{}M_1+b_{12}M_3+b_{21}M_3+b_{22}M_4</math>
<math>=b_{11}\left(\frac{1}{3}\left(N_1+N_2+N_3+N_4\right)-3N_1\right)+\ldots</math>

<math>=\mbox{ a linear combination of }N_1^{},N_2,N_3,N_4</math>
 

 
<math>\mbox{Proof of 3}{}_{}^{}</math>

<math>\mbox{Indeed in }a_1^{}M_1+a_2M_2+a_3M_3=
\begin{pmatrix}a_1&a_2\\a_3&0\end{pmatrix}\mbox{ lower right corner is always } 0
</math>

<math>\mbox{for example }\begin{pmatrix}240&157\\e&\pi\end{pmatrix}\mbox{ not in span.}</math>
 

 
<math>\mbox{Proof of 4}{}_{}^{}</math>

<math>a_1^{}N_1+a_2N_2+a_3N_3=\begin{pmatrix}a_2+a_3&a_1+a_3\\a_1+a_2&a_1+a_2+a_3\end{pmatrix}</math>

<math>\begin{pmatrix}240&157\\e&\pi\end{pmatrix}\mbox{ is equal? }
\begin{cases}240=a_2+a_3\\157=a_1+a_3\\e=a_1+a_2\\\pi=a_1+a_2+a_3\end{cases}\Rightarrow\mbox{No solution}</math>
 

 
<math>\mbox{Motivation}{}_{}^{}</math>

<math>S\subset V\mbox{ is linearly dependent if it is wasteful,}</math>
<math>\mbox{i.e. if }\exists v\in V\mbox{ such that }\exists a_1^{}\ldots a_n\in F \mbox{ and }u_1^{}\ldots u_2\in S</math>
<math>\mbox{ and }\exists b_1^{}\ldots b_m\in F \mbox{ and }w_1\ldots w_m\in S</math>

<math>\mbox{so that }\sum_{i=1}^na_iu_i=v=\sum_{i=1}^mb_iw_i</math>

<math>\sum a_iu_i-\sum b_iw_i=0</math>

<math>\mbox{can be represented as }\sum c_iz_i=0</math>
 

 
<math>\mbox{Definition}{}_{}^{}</math>

<math>S\subset V\mbox{ is called linearly dependent if you can find }</math>
<math>z_1^{}\ldots z_n\in S\mbox{ different from each other and }c_1^{}\ldots c_n\in F\mbox{ so that not all of which are } 0,</math>
<math>\mbox{so that }\sum c_iz_i=0
\mbox{ otherwise, }S\mbox{ is called linearly independent}</math>
 

 
<math>\mbox{Example 1}{}_{}^{}</math>

<math>\mbox{In }\mathbb{R}, S=
\lbrace\begin{pmatrix}1&2&3\end{pmatrix},
\begin{pmatrix}4&5&6\end{pmatrix},
\begin{pmatrix}7&8&9\end{pmatrix}\rbrace\mbox{ is linearly dependent}</math>

<math>1\cdot\begin{pmatrix}1&2&3\end{pmatrix}-
2\cdot\begin{pmatrix}4&5&6\end{pmatrix}+
1\cdot\begin{pmatrix}7&8&9\end{pmatrix}=0</math>
 

 
<math>\mbox{Example 2}{}_{}^{}</math>

<math>\mathbb{R}^n, e_i=\begin{pmatrix}0\\\vdots\\1\\\vdots\\0\end{pmatrix}i^{th}\mbox{ row}</math>

<math>S=\lbrace e_1^{},\ldots,e_n\rbrace</math>

<math>\mbox{Claim }S\mbox{ is linearly independent}{}_{}^{}</math>

<math>\begin{pmatrix}0\\\vdots\\0\end{pmatrix}=0
=\sum_{i=1}^na_ie_i
=\begin{pmatrix}a_1\\a_2\\\vdots\\a_n\end{pmatrix}\Rightarrow
\begin{matrix}a_1=0\\a_2=0\\\vdots\\a_n=0\end{matrix}</math>

<math>\mbox{not not all }a_i^{}\mbox{ are }0\Rightarrow \mbox{ not linearly dependent.}</math>

<math>\mbox{Claim }S\subset V\mbox{ is linearly independent iff whenever }\sum a_iu_i=0
\mbox{ and distinct }u_i\in S\mbox{ then }\forall i\quad a_i=0</math>
 

 
<math>\mbox{Comments}{}_{}^{}</math>

#<math>\emptyset\subset V\mbox{ is linearly independent}</math>
#<math>\mbox{Suppose }u\in V,\quad \lbrace u\rbrace\mbox{ the singleton set is linearly independent iff }u_{}^{}\neq 0</math>

<math>\lbrace0\rbrace\mbox{ is linearly dependent. example }7\cdot0=0</math>

<math>\mbox{if }u\neq0\mbox{ assume }a\cdot u=0\mbox{, and }a\neq0\Rightarrow a_{}^{-1}au=0
\Rightarrow u=0\mbox{ contradiction results, so no such }a\mbox{ exists.}</math>
<math>\mbox{ So}{}_{}^{}\lbrace u\rbrace\mbox{is not linearly dependent, hence it is linearly independent.}</math>