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MathGroup Archive 2006

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Re: Determining Linear dependent vectors

  • To: mathgroup at smc.vnet.net
  • Subject: [mg66261] Re: Determining Linear dependent vectors
  • From: bghiggins at ucdavis.edu
  • Date: Sat, 6 May 2006 01:54:32 -0400 (EDT)
  • References: <e3f5s6$sac$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

For example, if you have the vectors:
  v1={2,0,0,1}, v2={1,0,1,1},v3={0,4,0,0},v5={0,0,3,0},v6={0,0,0,2}

You can use RowReduce to determine the number of linearly inndependent
vectors in your set.


Write the vectors as rows of a matrix:

data = {{2, 0, 0, 1}, {1, 0, 1, 1}, {0, 4, 0, 0}, {0, 0, 3, 0}, {0, 0,
0, 2}}

RowrReduce[data]

{{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}, {0, 0, 0, 0}}

Thus a basis set for your system  is the non-zero rows shown above. If
you want to write your set of vectors in terms of this basis set, then
set up the data as column vectors and use RowReduce.

RowReduce[Transpose[data]]

{{1, 0, 0, 0, -2}, {0, 1, 0, 0, 4}, {0, 0, 1, 0, 0},
  {0, 0, 0, 1, -(4/3)}}

This shows that v5=-2v1+4v2-(4/3)v4

and

RowReduce[Transpose[data[[{1, 2, 3, 5, 4}]]]]

{{1, 0, 0, 0, -(3/2)}, {0, 1, 0, 0, 3}, {0, 0, 1, 0, 0},
  {0, 0, 0, 1, -(3/4)}}

This shows that

v4=-(3/2)v1+3v2+v3-(3/4)v5

and so forth.

Hope this helps,

Cheers,

Brian

Saurabh wrote:
> Am looking for methods to determine linearly dependent vectors out of a given set. Any pointers appreciated.
> 
> Thanks,


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