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{{06-240/Navigation}} |
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<math>\mbox{From last class}{}_{}^{}</math> |
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===Links to Classnotes=== |
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* PDF file by [[User:Alla]]: [[Media:MAT_Lect008.pdf|Week 4 Lecture 2 notes]] |
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* PDF file by [[User:Gokmen]]: [[Media:06-240-lecture-05-october.pdf|Week 4 Lecture 2 notes]] |
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===Scan of Tutorial notes=== |
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* PDF file by [[User:Alla]]: [[Media:MAT_Tut004.pdf|Week 4 Tutorial notes]] |
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----<math>\mbox{From last class}{}_{}^{}</math> |
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<math>M_1=\begin{pmatrix}1&0\\0&0\end{pmatrix}, |
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<math>M_1=\begin{pmatrix}1&0\\0&0\end{pmatrix}, |
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<math>\mbox{Examples}{}_{}^{}</math> |
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<math>\mbox{Examples}{}_{}^{}</math> |
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<math>1. \beta=\emptyset{}_{}^{}\mbox{ is a basis of }\lbrace0\rbrace</math> |
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<math>2. {}_{}^{}V\mbox{ be }\mathbb{R}\mbox{ as a vector space over }\mathbb{R}</math> |
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<math>\qquad{}_{}^{}\beta=\lbrace5\rbrace\mbox{ and }\beta=\lbrace1\rbrace\mbox{ are bases.}</math> |
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<math>3.{}_{}^{}\mbox{ Let }V\mbox{ be }\mathbb{C}\mbox{ as a vector space over }\mathbb{R} \quad\beta=\lbrace1,i\rbrace</math> |
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:<math>\qquad{}_{}^{}\mbox{Check}</math> |
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:<math>\qquad{}_{}^{}\mbox{1. Every complex number is a linear combination of }\beta.</math> |
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::<math>Z=a+bi=a\cdot 1+b\cdot i\mbox{ with coefficients in }\mathbb{R}\mbox{ so }\lbrace1,i\rbrace\mbox{ generates}</math> |
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:<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> |
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::<math>{}_{}^{}\Rightarrow a+bi=0\Rightarrow a=0\mbox{ and } b=0</math> |
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<math>{}_{}^{}\mbox{4. }V\in\mathbb{R}^n= |
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\left\lbrace\begin{pmatrix}\vdots\end{pmatrix}y,\qquad |
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e_1=\begin{pmatrix}1\\0\\\vdots\\0\end{pmatrix}, |
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e_2=\begin{pmatrix}0\\1\\\vdots\\0\end{pmatrix},\ldots, |
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e_n=\begin{pmatrix}0\\0\\\vdots\\1\end{pmatrix}\right\rbrace</math> |
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:<math>{}_{}^{}e_1\ldots e_n\mbox{ are a basis of }V</math> |
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::<math>{}_{}^{}\mbox{They span }\begin{pmatrix}a_1\\\vdots\\a_n\end{pmatrix}=\sum a_ie_i</math> |
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::<math>{}_{}^{}\mbox{They are linearly independent. }\sum a_ie_i=0\Rightarrow \sum a_ie_i= |
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\begin{pmatrix}a_1\\\vdots\\a_n\end{pmatrix}=0\Rightarrow a_i=0 \quad\forall i</math> |
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<math>{}_{}^{}\mbox{5. In }V=P_3(\mathbb{R}),\qquad \beta=\lbrace 1,x,x^2,x^3\rbrace</math> |
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<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> |
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:<math>{}_{}^{}\mbox{1. Generate }</math> |
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::<math>u_1+u_2=2\Rightarrow \frac{1}{2}(u_1+u_2)=1\mbox{ so }1 \in\mbox{ span }S</math> |
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::<math>u_1-u_2=2x\Rightarrow \frac{1}{2}(u_1-u_2)=x\mbox{ so }x \in\mbox{ span }S</math> |
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::<math>{}_{}^{}\mbox{ so span}\lbrace 1,x\rbrace \subset\mbox{ span }\beta</math> |
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:<math>{}_{}^{}\mbox{2. Linearly independent. Assume }au_1+bu_2=0</math> |
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::<math>\Rightarrow a(1+x)+b(1-x)=0\Rightarrow a+b+(a-b)x=0</math> |
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::<math>{}_{}^{}\Rightarrow a+b=0\mbox{ and }a-b=0</math> |
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::<math>(a+b)+(a-b)\Rightarrow 2a=0\Rightarrow a=0</math> |
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::<math>(a+b)-(a-b)\Rightarrow 2b=0\Rightarrow b=0</math> |
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<br> |
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<br> |
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#
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Week of...
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Notes and Links
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1
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Sep 11
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About, Tue, HW1, Putnam, Thu
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2
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Sep 18
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Tue, HW2, Thu
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3
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Sep 25
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Tue, HW3, Photo, Thu
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4
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Oct 2
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Tue, HW4, Thu
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5
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Oct 9
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Tue, HW5, Thu
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6
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Oct 16
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Why?, Iso, Tue, Thu
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7
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Oct 23
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Term Test, Thu (double)
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8
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Oct 30
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Tue, HW6, Thu
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9
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Nov 6
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Tue, HW7, Thu
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10
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Nov 13
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Tue, HW8, Thu
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11
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Nov 20
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Tue, HW9, Thu
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12
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Nov 27
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Tue, HW10, Thu
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13
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Dec 4
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On the final, Tue, Thu
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F
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Dec 11
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Final: Dec 13 2-5PM at BN3, Exam Forum
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Register of Good Deeds
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![06-240-ClassPhoto.jpg](/images/thumb/8/82/06-240-ClassPhoto.jpg/180px-06-240-ClassPhoto.jpg) Add your name / see who's in!
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edit the panel
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Links to Classnotes
Scan of Tutorial notes
![{\displaystyle \emptyset \subset V{\mbox{ is linearly independent}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2eaa82f8191774ed56c5b9190e041bf20b88a3a)
![{\displaystyle \lbrace u\rbrace {\mbox{ is linearly independent iff }}u_{}^{}\neq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19c90a0276de433f36413ae05491fb0160614b8d)
![{\displaystyle {\mbox{If }}S_{1}^{}{\mbox{ is linearly dependent, so is }}S_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be48182817a998a61f2b2300c15eb8c73422b8fe)
![{\displaystyle {\mbox{If }}S_{2}^{}{\mbox{ is linearly dependent, so is }}S_{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3bcec66c67db8412925bb8bbac3d5fbc4baee09)
![{\displaystyle {\mbox{If }}S_{1}^{}{\mbox{ generates }}V{\mbox{, so does }}S_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45dc88ee8eed74123960281713d2fa32590f1e7e)
![{\displaystyle {\mbox{If }}S_{2}^{}{\mbox{ does not generate }}V{\mbox{ neither does }}S_{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a59f6715b5ed278df4c559478c214628d6e7c642)
![{\displaystyle {\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.}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51787572af0bca35f9ac91f115a05d39122e0030)
![{\displaystyle {}_{}^{}\beta {\mbox{ generates }}V{\mbox{ or }}V={\mbox{ span }}\beta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8abd82cd8b56ed59e5860c5ee16417273650ebb4)
![{\displaystyle {}_{}^{}\beta {\mbox{ is linearly independent.}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4740809a9948d50ded122ee588602f46c2a1ce3)
![{\displaystyle \qquad {}_{}^{}{\mbox{Check}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f1e8048433714622fb3575092ddf87d52dd2ae3)
![{\displaystyle Z=a+bi=a\cdot 1+b\cdot i{\mbox{ with coefficients in }}\mathbb {R} {\mbox{ so }}\lbrace 1,i\rbrace {\mbox{ generates}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25714e5722541d218216f8f3818b180632d4c42b)
![{\displaystyle {}_{}^{}\Rightarrow a+bi=0\Rightarrow a=0{\mbox{ and }}b=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/149d3cce6feb0b610f891221fb2dffaeb346f483)
![{\displaystyle {}_{}^{}{\mbox{They span }}{\begin{pmatrix}a_{1}\\\vdots \\a_{n}\end{pmatrix}}=\sum a_{i}e_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29849e916aff5aab6b714fa5bb5350788d4bbcd5)
![{\displaystyle {}_{}^{}{\mbox{They are linearly independent. }}\sum a_{i}e_{i}=0\Rightarrow \sum a_{i}e_{i}={\begin{pmatrix}a_{1}\\\vdots \\a_{n}\end{pmatrix}}=0\Rightarrow a_{i}=0\quad \forall i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb90091b6ab91f9cdce3a442663c95c40c90763a)
![{\displaystyle u_{1}+u_{2}=2\Rightarrow {\frac {1}{2}}(u_{1}+u_{2})=1{\mbox{ so }}1\in {\mbox{ span }}S}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11519fa48e3aafa296ccaa944cf34663e505c0dd)
![{\displaystyle u_{1}-u_{2}=2x\Rightarrow {\frac {1}{2}}(u_{1}-u_{2})=x{\mbox{ so }}x\in {\mbox{ span }}S}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f641feee777bcbd3e02da9ce95bfd9abeee2a6ce)
![{\displaystyle {}_{}^{}{\mbox{ so span}}\lbrace 1,x\rbrace \subset {\mbox{ span }}\beta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c47e5abc8b84b66e0a54de47e3ff1a2c4e67867)
![{\displaystyle \Rightarrow a(1+x)+b(1-x)=0\Rightarrow a+b+(a-b)x=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c3dcd3824a787bb813403a71f79f08ff5af9c2d)
![{\displaystyle {}_{}^{}\Rightarrow a+b=0{\mbox{ and }}a-b=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3f79e9866fba90a35424c8bcdfb52e23175e812)
![{\displaystyle (a+b)+(a-b)\Rightarrow 2a=0\Rightarrow a=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0c518a67cf3152003c968e8265c3b0dfdbd52fd)
![{\displaystyle (a+b)-(a-b)\Rightarrow 2b=0\Rightarrow b=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d1c8035dcb497065527eee28e7b147c402c8bb8)
![{\displaystyle {}_{}^{}\beta {\mbox{ generates}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02bbd45862e483686c0194bee4994c474e716d6f)
![{\displaystyle {}_{}^{}\beta {\mbox{ is linearly independent}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6608be09a55663a4a0a7f8fa4f1e4818b02958ba)