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johndoe
 Post subject: Vector space
PostPosted: Mon, 16 Jul 2012 22:48:30 UTC 
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Question : Is the set Q of rational numbers a real vector space (i.e. scalars are the real
numbers)? Explain your answer.

Yes, since all rational numbers are real numbers?

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outermeasure
 Post subject: Re: Vector space
PostPosted: Tue, 17 Jul 2012 02:42:50 UTC 
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johndoe wrote:
Question : Is the set Q of rational numbers a real vector space (i.e. scalars are the real
numbers)? Explain your answer.

Yes, since all rational numbers are real numbers?


In \mathbb{Q}, can you multiply by, e.g. \sqrt{2}\in\mathbb{R}?

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\begin{aligned} Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\ Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\ Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\ Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\ Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\ Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4 \end{aligned}
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Shadow
 Post subject: Re: Vector space
PostPosted: Tue, 17 Jul 2012 03:47:31 UTC 
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johndoe wrote:
Question : Is the set Q of rational numbers a real vector space (i.e. scalars are the real
numbers)? Explain your answer.

Yes, since all rational numbers are real numbers?


You need that real times rational is rational...

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johndoe
 Post subject: Re: Vector space
PostPosted: Tue, 17 Jul 2012 17:02:31 UTC 
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outermeasure wrote:
johndoe wrote:
Question : Is the set Q of rational numbers a real vector space (i.e. scalars are the real
numbers)? Explain your answer.

Yes, since all rational numbers are real numbers?


In \mathbb{Q}, can you multiply by, e.g. \sqrt{2}\in\mathbb{R}?




o so no, since I get irrational numbers after certain axiom operations.

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Shadow
 Post subject: Re: Vector space
PostPosted: Wed, 18 Jul 2012 04:52:07 UTC 
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johndoe wrote:
outermeasure wrote:
johndoe wrote:
Question : Is the set Q of rational numbers a real vector space (i.e. scalars are the real
numbers)? Explain your answer.

Yes, since all rational numbers are real numbers?


In \mathbb{Q}, can you multiply by, e.g. \sqrt{2}\in\mathbb{R}?




o so no, since I get irrational numbers after certain axiom operations.


What axiom operations? You mean that there is no closure under the field action.

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johndoe
 Post subject: Re: Vector space
PostPosted: Wed, 18 Jul 2012 15:40:59 UTC 
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Shadow wrote:
johndoe wrote:
outermeasure wrote:
johndoe wrote:
Question : Is the set Q of rational numbers a real vector space (i.e. scalars are the real
numbers)? Explain your answer.

Yes, since all rational numbers are real numbers?


In \mathbb{Q}, can you multiply by, e.g. \sqrt{2}\in\mathbb{R}?




o so no, since I get irrational numbers after certain axiom operations.


What axiom operations? You mean that there is no closure under the field action.


That I mean I engage in all those associativity, commutativity.. properties.

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Shadow
 Post subject: Re: Vector space
PostPosted: Wed, 18 Jul 2012 18:49:37 UTC 
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johndoe wrote:
Shadow wrote:
johndoe wrote:
outermeasure wrote:
johndoe wrote:
Question : Is the set Q of rational numbers a real vector space (i.e. scalars are the real
numbers)? Explain your answer.

Yes, since all rational numbers are real numbers?


In \mathbb{Q}, can you multiply by, e.g. \sqrt{2}\in\mathbb{R}?




o so no, since I get irrational numbers after certain axiom operations.


What axiom operations? You mean that there is no closure under the field action.


That I mean I engage in all those associativity, commutativity.. properties.


I'm a little confused, why are you thinking about associativity and commutativity? Those are fine, \mathbb{Q}\subseteq\mathbb{R} and \mathbb{R} is a field, so those will all work fine, it's closure that's the issue.

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