06-240/Classnotes For Thursday October 5

${\mbox{From last class}}{}_{}^{}$

$M_{1}={\begin{pmatrix}1&0\\0&0\end{pmatrix}},M_{2}={\begin{pmatrix}0&1\\0&0\end{pmatrix}},M_{3}={\begin{pmatrix}0&0\\1&0\end{pmatrix}},M_{4}{\begin{pmatrix}0&0\\0&1\end{pmatrix}}$

$N_{1}={\begin{pmatrix}0&1\\1&1\end{pmatrix}},N_{2}={\begin{pmatrix}1&0\\1&1\end{pmatrix}},N_{3}={\begin{pmatrix}1&1\\0&1\end{pmatrix}},N_{4}{\begin{pmatrix}1&1\\1&0\end{pmatrix}}$

${\mbox{The }}M_{i}{\mbox{s generate }}M_{2\times 2}$

${\mbox{Fact }}T\subset {\mbox{ span }}S\Rightarrow {\mbox{ span }}T\subset {\mbox{ span }}S$

$S\subset V{\mbox{ is linearly independent }}\Leftrightarrow {\mbox{ whenever }}u_{i}\in S{\mbox{ are distinct}}$

$\sum a_{i}u_{i}=0\Rightarrow V_{i}a_{i}=0{\mbox{ waste not}}$

${\mbox{Comments}}{}_{}^{}$

1. $\emptyset \subset V{\mbox{ is linearly independent}}$
2. $\lbrace u\rbrace {\mbox{ is linearly independent iff }}u_{}^{}\neq 0$
3. ${\mbox{If }}S_{1}^{}\subset S_{2}\subset V$
   1. ${\mbox{If }}S_{1}^{}{\mbox{ is linearly dependent, so is }}S_{2}$
   2. ${\mbox{If }}S_{2}^{}{\mbox{ is linearly dependent, so is }}S_{1}$
   3. ${\mbox{If }}S_{1}^{}{\mbox{ generates }}V{\mbox{, so does }}S_{2}$
   4. ${\mbox{If }}S_{2}^{}{\mbox{ does not generate }}V{\mbox{ neither does }}S_{1}$
4. ${\mbox{If }}S_{}^{}{\mbox{ is linearly independent in }}V{\mbox{ and }}v\notin S{\mbox{ then }}S\cup \lbrace u\rbrace {\mbox{ is linearly independent.}}$

${\mbox{Proof}}{}_{}^{}$

${\mbox{1.}}\Leftarrow :{\mbox{ start from second assertion and deduce first.}}$

${\mbox{Assume }}v_{}^{}\in {\mbox{span }}S$ $v=\sum a_{i}u_{i}{\mbox{ where }}u_{i}\in S,a_{i}\in F$

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

$(*)\qquad \sum a_{i}u_{i}+bv=0{\mbox{ where }}u_{i}\in S{\mbox{ and not all of the }}a_{i}{\mbox{ and }}b{\mbox{ are }}0$

${\mbox{If }}b=0{\mbox{, then }}\sum a_{i}u_{i}=0{\mbox{ and not }}a_{i}{\mbox{s are }}0$ ${}_{}^{}\Rightarrow S{\mbox{ is linearly dependent}}$ ${}_{}^{}{\mbox{but initial assumption was }}S{\mbox{ is linearly independent.}}\Rightarrow {\mbox{ contradiction so }}b\neq 0$ ${\mbox{So divide by }}b{\mbox{: (*) becomes }}\sum {\frac {a_{i}}{b}}u_{i}+v=0\Rightarrow v=-\sum {\frac {a_{i}}{b}}u_{i}\Rightarrow v\in {\mbox{ span }}S$

${\mbox{Definition}}{}_{}^{}$

${}_{}^{}{\mbox{A basis of a vector space }}V{\mbox{ is a subset }}\beta \subset V$ ${}_{}^{}{\mbox{such that}}$

1. ${}_{}^{}\beta {\mbox{ generates }}V{\mbox{ or }}V={\mbox{ span }}\beta$
2. ${}_{}^{}\beta {\mbox{ is linearly independent.}}$

${\mbox{Examples}}{}_{}^{}$

$1.\beta =\emptyset {}_{}^{}{\mbox{ is a basis of }}\lbrace 0\rbrace$

$2.{}_{}^{}V{\mbox{ be }}\mathbb {R} {\mbox{ as a vector space over }}\mathbb {R}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \qquad{}_{}^{}\beta=\lbrace5\rbrace\mbox{ and }\beta=\lbrace1\rbrace\mbox{ are bases.}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3.{}_{}^{}\mbox{ Let }V\mbox{ be }\mathbb{C}\mbox{ as a vector space over }\mathbb{R} \quad\beta=\lbrace1,i\rbrace}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \qquad{}_{}^{}\mbox{Check}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \qquad{}_{}^{}\mbox{1. Every complex number is a linear combination of }\beta.}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z=a+bi=a\cdot 1+b\cdot i\mbox{ with coefficients in }\mathbb{R}\mbox{ so }\lbrace1,i\rbrace\mbox{ generates}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {}_{}^{}\Rightarrow a+bi=0\Rightarrow a=0\mbox{ and } b=0}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {}_{}^{}\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}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {}_{}^{}e_1\ldots e_n\mbox{ are a basis of }V}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {}_{}^{}\mbox{They span }\begin{pmatrix}a_1\\\vdots\\a_n\end{pmatrix}=\sum a_ie_i}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {}_{}^{}\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}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {}_{}^{}\mbox{5. In }V=P_3(\mathbb{R}),\qquad \beta=\lbrace 1,x,x^2,x^3\rbrace}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {}_{}^{}\mbox{6. In }V=P_1(\mathbb{R})=\lbrace ax+b\rbrace,\qquad \beta=\lbrace 1+x,1-x\rbrace\mbox{ is a basis}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {}_{}^{}\mbox{1. Generate }}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_1+u_2=2\Rightarrow \frac{1}{2}(u_1+u_2)=1\mbox{ so }1 \in\mbox{ span }S}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_1-u_2=2x\Rightarrow \frac{1}{2}(u_1-u_2)=x\mbox{ so }x \in\mbox{ span }S}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {}_{}^{}\mbox{ so span}\lbrace 1,x\rbrace \subset\mbox{ span }\beta}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {}_{}^{}\mbox{2. Linearly independent. Assume }au_1+bu_2=0}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Rightarrow a(1+x)+b(1-x)=0\Rightarrow a+b+(a-b)x=0}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {}_{}^{}\Rightarrow a+b=0\mbox{ and }a-b=0}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a+b)+(a-b)\Rightarrow 2a=0\Rightarrow a=0}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a+b)-(a-b)\Rightarrow 2b=0\Rightarrow b=0}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mbox{Theorem}{}_{}^{}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {}_{}^{}\mbox{A subset }\beta\mbox{ of a vectorspace }V \mbox{ is a basis iff every }v\in V\mbox{ can be expressed as}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {}_{}^{}\mbox{a linear combination of elements in }} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {}_{}^{}\beta \mbox{ in exactly one way.}}

${\mbox{Proof}}{}_{}^{}$

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {}_{}^{}\mbox{It is a combination of things we already know.}}

1. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {}_{}^{}\beta\mbox{ generates}}
2. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {}_{}^{}\beta\mbox{ is linearly independent}}