06-240/Classnotes For Tuesday October 3

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

$v\in V{\mbox{ is a linear combination of elements in }}S\subset V$ ${\mbox{ if }}\exists u_{1},\ldots ,u_{n}\in S{\mbox{ and }}a_{1},\dots ,a_{n}\in F{\mbox{ such that }}V=\sum a_{i}u_{i}$

${\mbox{Example}}{}_{}^{}$

${\mbox{In }}P_{3}(\mathbb {R} ){\mbox{,}}$ $v_{1}^{}=2x^{3}-2x^{2}+12-6{\mbox{ is a linear combination of:}}$ $u_{1}^{}=x^{3}-2x^{2}-5x-3{\mbox{ and }}u_{2}=3x^{3}-5x^{2}-4x-9$ ${\mbox{but }}v_{2}^{}=3x^{3}-2x^{2}+7x+8{\mbox{ is not.}}$

${\mbox{Why?}}{}_{}^{}$

$v_{1}^{}=2x^{3}-2x^{2}+12-6=a_{1}^{}u_{1}+a_{2}u_{2}=a_{1}(x^{3}-2x^{2}-5x-3)+a_{2}(3x^{3}-5x^{2}-4x-9)$ $v_{1}^{}=-4u_{1}+2u_{2}$

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

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

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

$V=M_{2\times 2}(\mathbb {R} )$

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

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

${\mbox{Claims}}{}_{}^{}$

1. $\lbrace M_{1}^{},M_{2},M_{3},M_{4}\rbrace {\mbox{ generates }}V$
2. $\lbrace N_{1}^{},N_{2},N_{3},N_{4}\rbrace {\mbox{ generates }}V$
3. $\lbrace M_{1}^{},M_{2},M_{3}\rbrace {\mbox{ does not generate }}V$
4. $\lbrace N_{1}^{},N_{2},N_{3}\rbrace {\mbox{ does not generate }}V$

${\mbox{Proof of 1}}{}_{}^{}$

${\mbox{Given any }}B={\begin{pmatrix}b_{11}^{}&b_{12}\\b_{21}&b_{22}\end{pmatrix}}{\mbox{ need to find }}a_{1},a_{2},a_{3},a_{4}{\mbox{ such that,}}$

${\begin{pmatrix}b_{11}^{}&b_{12}\\b_{21}&b_{22}\end{pmatrix}}=B=a_{1}M_{1}+a_{2}M_{2}+a_{3}M_{3}+a_{4}M_{4}={\begin{pmatrix}a_{1}&0\\0&0\end{pmatrix}}+{\begin{pmatrix}0&a_{2}\\0&0\end{pmatrix}}+{\begin{pmatrix}0&0\\a_{3}&0\end{pmatrix}}+{\begin{pmatrix}0&0\\0&a_{4}\end{pmatrix}}$

$={\begin{pmatrix}a_{1}^{}&a_{2}\\a_{3}&a_{4}\end{pmatrix}}\Leftrightarrow {\begin{cases}b_{11}=a_{1}\\b_{12}=a_{2}\\b_{21}=a_{3}\\b_{22}=a_{4}\end{cases}}$ ${\mbox{A system of 4 equations with 4 unknowns}}{}_{}^{}$
${\mbox{Proof of 2}}{}_{}^{}$

${\begin{pmatrix}b_{11}^{}&b_{12}\\b_{21}&b_{22}\end{pmatrix}}=B=a_{1}N_{1}+a_{2}N_{2}+a_{3}N_{3}+a_{4}N_{4}={\begin{pmatrix}0&a_{1}\\a_{1}&a_{1}\end{pmatrix}}+{\begin{pmatrix}a_{2}&0\\a_{2}&a_{2}\end{pmatrix}}+{\begin{pmatrix}a_{3}&a_{3}\\0&a_{3}\end{pmatrix}}+{\begin{pmatrix}a_{4}&a_{4}\\a_{4}&0\end{pmatrix}}$

$={\begin{pmatrix}a_{2}^{}+a_{3}+a_{4}&a_{1}+a_{3}+a_{4}\\a_{1}+a_{2}+a_{4}&a_{1}+a_{2}+a_{3}\end{pmatrix}}\Leftrightarrow {\begin{cases}b_{11}=a_{2}+a_{3}+a_{4}\\b_{12}=a_{1}+a_{3}+a_{4}\\b_{21}=a_{1}+a_{2}+a_{4}\\b_{22}=a_{1}+a_{2}+a_{3}\end{cases}}$

${\mbox{Trick}}{}_{}^{}$

$M_{1}={\frac {1}{3}}\left(N_{1}+N_{2}+N_{3}+N_{4}\right)-3N_{1}$ $M_{2}={\frac {1}{3}}\left(N_{1}+N_{2}+N_{3}+N_{4}\right)-3N_{2}$ $M_{3}={\frac {1}{3}}\left(N_{1}+N_{2}+N_{3}+N_{4}\right)-3N_{3}$ $M_{4}={\frac {1}{3}}\left(N_{1}+N_{2}+N_{3}+N_{4}\right)-3N_{4}$

$B=b_{11}^{}M_{1}+b_{12}M_{3}+b_{21}M_{3}+b_{22}M_{4}$ 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 =b_{11}\left(\frac{1}{3}\left(N_1+N_2+N_3+N_4\right)-3N_1\right)+\ldots}

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 }N_1^{},N_2,N_3,N_4}

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{Proof of 3}{}_{}^{}}

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{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 }

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

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{Proof of 4}{}_{}^{}}

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_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}}

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 \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}}

${\mbox{Motivation}}{}_{}^{}$

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 S\subset V\mbox{ is linearly dependent if it is wasteful,}} 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{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} 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{ and }\exists b_1^{}\ldots b_m\in F \mbox{ and }w_1\ldots w_m\in 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 that }\sum_{i=1}^na_iu_i=v=\sum_{i=1}^mb_iw_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 \sum a_iu_i-\sum b_iw_i=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{can be represented as }\sum c_iz_i=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{Definition}{}_{}^{}}

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 S\subset V\mbox{ is called linearly dependent if you can find }} 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_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,} 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 that }\sum c_iz_i=0 \mbox{ otherwise, }S\mbox{ is called linearly independent}}

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{Example 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 \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}}

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 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}

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

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 S=\lbrace e_1^{},\ldots,e_n\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{Claim }S\mbox{ is linearly independent}{}_{}^{}}

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 \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}}

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{not not all }a_i^{}\mbox{ are }0\Rightarrow \mbox{ not linearly dependent.}}

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{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}

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{Comments}{}_{}^{}}

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 \emptyset\subset V\mbox{ is linearly independent}}
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 \mbox{Suppose }u\in V,\quad \lbrace u\rbrace\mbox{ the singleton set is linearly independent iff }u_{}^{}\neq 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 \lbrace0\rbrace\mbox{ is linearly dependent. example }7\cdot0=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{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.}} 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}{}_{}^{}\lbrace u\rbrace\mbox{is not linearly dependent, hence it is linearly independent.}}