Theoretical Physics Theoretical Physics Help Forum 
Nov 24th 2013, 06:25 AM

#1  Junior Member
Join Date: Nov 2013 Location: Chennai
Posts: 2
 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=TV), 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 (r1,r2,r3,...,rn) have been replaced by a single vector drawn in a space spanned by N linearly independent vectors (ri(q1,q2,q3,...,qn)) where q1,q2,q3,...,qn are the coordinates of the terminating end of the single vector in the Ntuple 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

 
Nov 24th 2013, 10:58 AM

#2  Forum Admin
Join Date: Apr 2008 Location: On the dance floor, baby!
Posts: 2,466

Originally Posted by Vivian 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=TV), 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 (r1,r2,r3,...,rn) have been replaced by a single vector drawn in a space spanned by N linearly independent vectors (ri(q1,q2,q3,...,qn)) where q1,q2,q3,...,qn are the coordinates of the terminating end of the single vector in the Ntuple 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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Nov 26th 2013, 06:00 AM

#3  Senior Member
Join Date: Apr 2008 Location: Bedford, England
Posts: 668
 Multidimensional visualisation
As Dan indicates, our visualisation processes are based on our natural everyday experience of the world.
By extrapolating from 2D via 3D one can perhaps gain a vague, dimly appreciated, visualisation of 4 "spacetype" 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 nonspacially, via colour or intensity (or texture, etc.)
But, often the best way is to simply trust the Maths...

 
Dec 1st 2013, 02:17 AM

#4  Junior Member
Join Date: Nov 2013 Location: Chennai
Posts: 2

Thank you Dan & MBW, really appreciated. Can you suggest a text book that helps well to deal with understanding vector spaces?
Regards,
Vivian.

 
Dec 1st 2013, 04:50 AM

#5  Forum Admin
Join Date: Apr 2008 Location: On the dance floor, baby!
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Originally Posted by Vivian 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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Dec 7th 2013, 07:25 AM

#6  Physics Team
Join Date: Apr 2009 Location: Boston's North Shore
Posts: 1,570

Originally Posted by Vivian 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://bookosz1.org/book/446177/c50aed 
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