06-240/Classnotes For Thursday October 5 - Revision history

Wongpak at 02:09, 25 October 2006

2006-10-25T02:09:38Z

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			{{06-240/Navigation}}

	===Links to Classnotes===		===Links to Classnotes===

Wongpak: /* Links to Classnotes */

2006-10-25T02:08:14Z

Links to Classnotes

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	* PDF file by [[User:Alla]]: [[Media:MAT_Lect008.pdf\|Week 4 Lecture 2 notes]]		* PDF file by [[User:Alla]]: [[Media:MAT_Lect008.pdf\|Week 4 Lecture 2 notes]]
			* PDF file by [[User:Gokmen]]: [[Media:06-240-lecture-05-october.pdf\|Week 4 Lecture 2 notes]]

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

Alla: Added scan of Week 4 Tutorial notes

2006-10-08T21:56:08Z

Added scan of Week 4 Tutorial notes

← Older revision		Revision as of 17:56, 8 October 2006
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	===Links to Classnotes===		===Links to Classnotes===

	* PDF file by [[User:Alla]]: [[Media:MAT_Lect008.pdf\|Week 4 Lecture 2 notes]]		* PDF file by [[User:Alla]]: [[Media:MAT_Lect008.pdf\|Week 4 Lecture 2 notes]]

			===Scan of Tutorial notes===

			* PDF file by [[User:Alla]]: [[Media:MAT_Tut004.pdf\|Week 4 Tutorial notes]]

	----<math>\mbox{From last class}{}_{}^{}</math>		----<math>\mbox{From last class}{}_{}^{}</math>

Alla: Added scan of Week 4 Lecture 2 notes

2006-10-08T21:52:34Z

Added scan of Week 4 Lecture 2 notes

← Older revision		Revision as of 17:52, 8 October 2006
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			===Links to Classnotes===
⚫	<math>\mbox{From last class}{}_{}^{}</math>
			* PDF file by [[User:Alla]]: [[Media:MAT_Lect008.pdf\|Week 4 Lecture 2 notes]]
		⚫	----<math>\mbox{From last class}{}_{}^{}</math>

	<math>M_1=\begin{pmatrix}1&0\\0&0\end{pmatrix},		<math>M_1=\begin{pmatrix}1&0\\0&0\end{pmatrix},

Mc: Add missing examples

2006-10-08T00:54:47Z

Add missing examples

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

			<math>1. \beta=\emptyset{}_{}^{}\mbox{ is a basis of }\lbrace0\rbrace</math>

			<math>2. {}_{}^{}V\mbox{ be }\mathbb{R}\mbox{ as a vector space over }\mathbb{R}</math>
			<math>\qquad{}_{}^{}\beta=\lbrace5\rbrace\mbox{ and }\beta=\lbrace1\rbrace\mbox{ are bases.}</math>

			<math>3.{}_{}^{}\mbox{ Let }V\mbox{ be }\mathbb{C}\mbox{ as a vector space over }\mathbb{R} \quad\beta=\lbrace1,i\rbrace</math>

			:<math>\qquad{}_{}^{}\mbox{Check}</math>

			:<math>\qquad{}_{}^{}\mbox{1. Every complex number is a linear combination of }\beta.</math>
			::<math>Z=a+bi=a\cdot 1+b\cdot i\mbox{ with coefficients in }\mathbb{R}\mbox{ so }\lbrace1,i\rbrace\mbox{ generates}</math>

			:<math>\qquad{}_{}^{}\mbox{2. Show }\beta=\lbrace1,i\rbrace\mbox{ are linearly independent. Assume }a\cdot 1+b\cdot i=0\mbox{ where }a,b\in\mathbb{R}</math>
			::<math>{}_{}^{}\Rightarrow a+bi=0\Rightarrow a=0\mbox{ and } b=0</math>

			<math>{}_{}^{}\mbox{4. }V\in\mathbb{R}^n=
			\left\lbrace\begin{pmatrix}\vdots\end{pmatrix}y,\qquad
			e_1=\begin{pmatrix}1\\0\\\vdots\\0\end{pmatrix},
			e_2=\begin{pmatrix}0\\1\\\vdots\\0\end{pmatrix},\ldots,
			e_n=\begin{pmatrix}0\\0\\\vdots\\1\end{pmatrix}\right\rbrace</math>

			:<math>{}_{}^{}e_1\ldots e_n\mbox{ are a basis of }V</math>
			::<math>{}_{}^{}\mbox{They span }\begin{pmatrix}a_1\\\vdots\\a_n\end{pmatrix}=\sum a_ie_i</math>
			::<math>{}_{}^{}\mbox{They are linearly independent. }\sum a_ie_i=0\Rightarrow \sum a_ie_i=
			\begin{pmatrix}a_1\\\vdots\\a_n\end{pmatrix}=0\Rightarrow a_i=0 \quad\forall i</math>

			<math>{}_{}^{}\mbox{5. In }V=P_3(\mathbb{R}),\qquad \beta=\lbrace 1,x,x^2,x^3\rbrace</math>

			<math>{}_{}^{}\mbox{6. In }V=P_1(\mathbb{R})=\lbrace ax+b\rbrace,\qquad \beta=\lbrace 1+x,1-x\rbrace\mbox{ is a basis}</math>
			:<math>{}_{}^{}\mbox{1. Generate }</math>
			::<math>u_1+u_2=2\Rightarrow \frac{1}{2}(u_1+u_2)=1\mbox{ so }1 \in\mbox{ span }S</math>
			::<math>u_1-u_2=2x\Rightarrow \frac{1}{2}(u_1-u_2)=x\mbox{ so }x \in\mbox{ span }S</math>
			::<math>{}_{}^{}\mbox{ so span}\lbrace 1,x\rbrace \subset\mbox{ span }\beta</math>
			:<math>{}_{}^{}\mbox{2. Linearly independent. Assume }au_1+bu_2=0</math>
			::<math>\Rightarrow a(1+x)+b(1-x)=0\Rightarrow a+b+(a-b)x=0</math>
			::<math>{}_{}^{}\Rightarrow a+b=0\mbox{ and }a-b=0</math>
			::<math>(a+b)+(a-b)\Rightarrow 2a=0\Rightarrow a=0</math>
			::<math>(a+b)-(a-b)\Rightarrow 2b=0\Rightarrow b=0</math>
	<br>		<br>

Mc: Added Oct 5 Lecture Notes

2006-10-07T23:34:43Z

Added Oct 5 Lecture Notes

New page

<math>\mbox{From last class}{}_{}^{}</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{The }M_i\mbox{s generate }M_{2\times 2}</math>

<math>\mbox{Fact }T\subset\mbox{ span }S\Rightarrow \mbox{ span }T\subset\mbox{ span }S </math>

<math>S\subset V\mbox{ is linearly independent }\Leftrightarrow \mbox{ whenever }u_i\in S\mbox{ are distinct}</math>

<math>\sum a_iu_i=0\Rightarrow V_ia_i=0 \mbox{ waste not}</math>
 

 
<math>\mbox{Comments}{}_{}^{}</math>
#<math>\emptyset\subset V\mbox{ is linearly independent}</math>
#<math>\lbrace u\rbrace\mbox{ is linearly independent iff }u_{}^{}\neq 0</math>
#<math>\mbox{If }S_1^{}\subset S_2\subset V</math>
##<math>\mbox{If }S_1^{}\mbox{ is linearly dependent, so is }S_2</math>
##<math>\mbox{If }S_2^{}\mbox{ is linearly dependent, so is }S_1</math>
##<math>\mbox{If }S_1^{}\mbox{ generates }V\mbox{, so does }S_2</math>
##<math>\mbox{If }S_2^{}\mbox{ does not generate }V\mbox{ neither does }S_1</math>
#<math>\mbox{If }S_{}^{}\mbox{ is linearly independent in }V\mbox{ and }v\notin S\mbox{ then }S\cup\lbrace u\rbrace\mbox{ is linearly independent.}</math>
 

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

<math>\mbox{1.}\Leftarrow:\mbox{ start from second assertion and deduce first.}</math>

<math>\mbox{Assume }v_{}^{}\in \mbox{span }S</math>
<math>v=\sum a_iu_i\mbox{ where }u_i\in S, a_i\in F</math>

<math>\sum a_iu_i-1\cdot v=0\mbox{ this is a linear combination of elements in }S\cup v</math>
<math>\mbox{ in which not all coefficients are }0 \mbox{ and which add to }0_{}^{}.</math>
<math>\mbox{So }S\cup \lbrace v\rbrace\mbox{ is linearly dependent by definition}</math>
 
<math>\mbox{2.}:\Rightarrow\mbox{ Assume }S\cup \lbrace v\rbrace\mbox{ is linearly dependent }\Rightarrow\mbox{ a linear combination can be found, of the form:}</math>

<math>(*)\qquad\sum a_iu_i+bv=0\mbox{ where }u_i\in S\mbox{ and not all of the }a_i \mbox{ and }b \mbox{ are }0</math>

<math>\mbox{If }b=0\mbox{, then }\sum a_iu_i=0\mbox{ and not }a_i\mbox{s are }0 </math>
<math>{}_{}^{}\Rightarrow S \mbox{ is linearly dependent}</math>
<math>{}_{}^{}\mbox{but initial assumption was }S\mbox{ is linearly independent.}\Rightarrow \mbox{ contradiction so }b\neq0</math>
<math>\mbox{So divide by }b\mbox{: (*) becomes }\sum\frac{a_i}{b}u_i + v = 0\Rightarrow v=-\sum\frac{a_i}{b}u_i\Rightarrow v\in \mbox{ span }S</math>
 

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

<math>{}_{}^{}\mbox{A basis of a vector space }V\mbox{ is a subset }\beta\subset V</math>
<math>{}_{}^{}\mbox{such that}</math>
#<math>{}_{}^{}\beta\mbox{ generates }V\mbox{ or }V=\mbox{ span }\beta</math>
#<math>{}_{}^{}\beta\mbox{ is linearly independent.}</math>
 

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

 

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

<math>{}_{}^{}\mbox{A subset }\beta\mbox{ of a vectorspace }V \mbox{ is a basis iff every }v\in V\mbox{ can be expressed as}</math>
<math>{}_{}^{}\mbox{a linear combination of elements in }</math>
<math>{}_{}^{}\beta \mbox{ in exactly one way.}</math>
 

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

<math>{}_{}^{}\mbox{It is a combination of things we already know.}</math>
#<math>{}_{}^{}\beta\mbox{ generates}</math>
#<math>{}_{}^{}\beta\mbox{ is linearly independent}</math>