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# can be multiplied by a number (not another vector) |
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Let <math>\mathcal F</math> be a field. A vector space <math>\mathbf V</math> over the field <math>\mathcal F</math> is a set <math>\mathbf V</math> (of vectors) with a special element <math>0_V</math>, a binary operation <math>+ : \mathbf V \times \mathbf V \rightarrow \mathbf V</math>, a binary operation <math>\cdot : \mathcal F \times \mathbf V \rightarrow \mathbf V</math>. |
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Let ''F'' be a field. A vector space ''V'' over the field ''F'' is a set ''V'' (of vectors) with a special element 0<sub>''V''</sub>, a binary operation + : ''V'' × ''V'' → ''V'', a binary operation • : ''F'' × ''V'' → ''V''. |
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{| style="border: solid 1px black" |
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| Convention for today: |
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| Convention for today: |
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: <math>x, y, z \in \mathbf V</math> |
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: <math>x, y, z \in \mathbf V</math> |
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: <math>a, b, c \in \mathcal F</math> |
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: <math>a, b, c \in F</math> |
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Latest revision as of 17:49, 24 September 2009
Additions to the MAT 240 web site no longer count towards good deed points
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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 7
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Tue, About, Thu
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2
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Sep 14
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Tue, HW1, HW1 Solution, Thu
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3
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Sep 21
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Tue, HW2, HW2 Solution, Thu, Photo
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4
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Sep 28
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Tue, HW3, HW3 Solution, Thu
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5
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Oct 5
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Tue, HW4, HW4 Solution, Thu,
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6
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Oct 12
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Tue, Thu
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7
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Oct 19
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Tue, HW5, HW5 Solution, Term Test on Thu
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8
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Oct 26
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Tue, Why LinAlg?, HW6, HW6 Solution, Thu
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9
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Nov 2
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Tue, MIT LinAlg, Thu
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10
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Nov 9
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Tue, HW7, HW7 Solution Thu
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11
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Nov 16
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Tue, HW8, HW8 Solution, Thu
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12
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Nov 23
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Tue, HW9, HW9 Solution, Thu
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13
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Nov 30
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Tue, On the final, Thu
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S
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Dec 7
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Office Hours
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F
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Dec 14
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Final on Dec 16
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To Do List
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The Algebra Song!
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Register of Good Deeds
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Misplaced Material
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![09-240-ClassPhoto.jpg](/images/thumb/6/6f/09-240-ClassPhoto.jpg/180px-09-240-ClassPhoto.jpg) Add your name / see who's in!
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Some links
WARNING: The notes below, written for students and by students, are provided "as is", with absolutely no warranty. They can not be assumed to be complete, correct, reliable or relevant. If you don't like them, don't read them. It is a bad idea to stop taking your own notes thinking that these notes can be a total replacement - there's nothing like one's own handwriting!
Visit this pages' history tab to see who added what and when.
Class notes for today
Vectors:
- can be added
- can be multiplied by a number (not another vector)
Let F be a field. A vector space V over the field F is a set V (of vectors) with a special element 0V, a binary operation + : V × V → V, a binary operation • : F × V → V.
Convention for today:
![{\displaystyle x,y,z\in \mathbf {V} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/269bd34abdf70e6c434c711598054841cc522367)
![{\displaystyle a,b,c\in F}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86f7db9e8e49083ab869efa4bc15b80ab0231d32)
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VS1 ![{\displaystyle \forall x,y\in \mathbf {V} ,x+y=y+x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29bdcebec7299be763479e032b8bbda761620b6c)
VS2 ![{\displaystyle \cdots (x+y)+z=x+(y+z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63c43a850794d63abbed19b84eed3d3c1d9b0437)
VS3 ![{\displaystyle \cdots x+0=x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/235e5bf69bd46422646b4f6ce46fe18e30371208)
VS4 ![{\displaystyle \forall x,\exists y{\mbox{ s.t. }}x+y=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2451e14bfb0b25103bcb6cdfdf076878d7567833)
VS5 ![{\displaystyle 1\cdot x=x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a768213d5d610701295b0387f4b80c1bcf070176)
VS6 ![{\displaystyle a\cdot (b\cdot x)=(a\cdot b)\cdot x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef127650592e646fe8e967e2250bfcaa3804f6de)
VS7 ![{\displaystyle a\cdot (x+y)=ax+ay}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f5f549b3a968fa6736849fa7f0e5437543e6607)
VS8
Proof of VS4
Take an arbitrary
Set
and note
![{\displaystyle x+y={\begin{pmatrix}a_{1}\\a_{2}\\\vdots \\a_{n}\end{pmatrix}}+{\begin{pmatrix}-a_{1}\\-a_{2}\\\vdots \\-a_{n}\end{pmatrix}}={\begin{pmatrix}a_{1}+(-a_{1})\\a_{2}+(-a_{2})\\\vdots \\a_{n}+(-a_{n})\end{pmatrix}}={\begin{pmatrix}0\\0\\\vdots \\0\end{pmatrix}}=0_{F^{n}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8c9813be2c677d80aafab0a2c3b6b826a8e9753)
Examples
![{\displaystyle F^{n}=\left\{{\begin{pmatrix}a_{1}\\a_{2}\\\vdots \\a_{n}\end{pmatrix}}:a_{i}\in F\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08b3895ef7d6ab94b6eaec49e4fed24dfa5ae53b)
![{\displaystyle {\begin{pmatrix}a_{1}\\a_{2}\\\vdots \\a_{n}\end{pmatrix}}+{\begin{pmatrix}b_{1}\\b_{2}\\\vdots \\b_{n}\end{pmatrix}}={\begin{pmatrix}a_{1}+b_{1}\\a_{2}+b_{2}\\\vdots \\a_{n}+b_{n}\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3598c3b2cbff2dcf9547e991f6245090e7a3685a)
![{\displaystyle a{\begin{pmatrix}b_{1}\\b_{2}\\\vdots \\b_{n}\end{pmatrix}}={\begin{pmatrix}ab_{1}\\ab_{2}\\\vdots \\ab_{n}\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38219a9f97ad565c61024179bfc86e679b1ca2b9)
- ...
- ...
![{\displaystyle {\mathcal {F}}(S,F)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2485fb4d0e674a22f0bcdbe8da1346fcb7c1a5e)
- Polynomials
![{\displaystyle ...}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d00f2f395950e3698a46501d1e9aae8e8defa145)
Food for thought
What is wrong with setting
- Unnecessary for a V.S.
- This is useless, since it does not describe reality. For example, a mathematical theory with 46 dimensions can be perfect and mathematically elegant, but if the only solution to it is a universe in which life cannot form it is not reality, hence we have no use for it.