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Old Nov 24th 2013, 06:25 AM   #1
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Smile Viewing vectors

Hello everybody, My name is Vivian. I have been going through vector spaces for quiet long now, but somehow I am unable to visualize a space spanned by N independent vectors. Can someone help me understand it with a simple practical example? More over I have been going through Lagrange's formulations in dynamics (L=T-V), where in I learnt that, N vectors viz. (r1,r2,r3,...,rn) have been replaced by a single vector drawn in a space spanned by N linearly independent vectors (ri(q1,q2,q3,...,qn)) where q1,q2,q3,...,qn are the co-ordinates of the terminating end of the single vector in the N-tuple space under consideration. Is my understanding correct about the Lagrange's formulation technique...??? Kindly help.

Thanks in advance.

Last edited by Vivian; Nov 24th 2013 at 06:30 AM. Reason: Grammatical errors
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Old Nov 24th 2013, 10:58 AM   #2
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Originally Posted by Vivian View Post
Hello everybody, My name is Vivian. I have been going through vector spaces for quiet long now, but somehow I am unable to visualize a space spanned by N independent vectors. Can someone help me understand it with a simple practical example? More over I have been going through Lagrange's formulations in dynamics (L=T-V), where in I learnt that, N vectors viz. (r1,r2,r3,...,rn) have been replaced by a single vector drawn in a space spanned by N linearly independent vectors (ri(q1,q2,q3,...,qn)) where q1,q2,q3,...,qn are the co-ordinates of the terminating end of the single vector in the N-tuple space under consideration. Is my understanding correct about the Lagrange's formulation technique...??? Kindly help.

Thanks in advance.
The only way I know how to visualize N dimensions is to try to look at the problem in 3 dimensions and try to draw some comparisons. (I suck at visualization.) Your analysis for the Lagrange method is correct.

-Dan
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Old Nov 26th 2013, 06:00 AM   #3
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Multidimensional visualisation

As Dan indicates, our visualisation processes are based on our natural every-day experience of the world.
By extrapolating from 2D via 3D one can perhaps gain a vague, dimly appreciated, visualisation of 4 "space-type" dimensions.
I generally consider myself quite adept at visualisation, but anything beyond 4D leaves my mind well and trully boggled.

One can sometimes usefully visualise other dimensions non-spacially, via colour or intensity (or texture, etc.)

But, often the best way is to simply trust the Maths...
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Old Dec 1st 2013, 02:17 AM   #4
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Thank you Dan & MBW, really appreciated. Can you suggest a text book that helps well to deal with understanding vector spaces?

Regards,
Vivian.
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Old Dec 1st 2013, 04:50 AM   #5
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Originally Posted by Vivian View Post
Thank you Dan & MBW, really appreciated. Can you suggest a text book that helps well to deal with understanding vector spaces?

Regards,
Vivian.
Depending on what you already know any Linear Algebra text will do you fine. If you've already done that then I'd recommend a Group theory text.

-Dan
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Old Dec 7th 2013, 07:25 AM   #6
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Originally Posted by Vivian View Post
Thank you Dan & MBW, really appreciated. Can you suggest a text book that helps well to deal with understanding vector spaces?

Regards,
Vivian.
Most texts on Linear Algebra are good enough for such a task but if you want a good text that covers everything try Mathematical Methods in the Physical Sciences by Mary Boas. You can get a copy of this version online at http://bookos-z1.org/book/446177/c50aed
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