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MathGroup Archive 2006

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Re: Re: Determining Linear dependent vectors

  • To: mathgroup at smc.vnet.net
  • Subject: [mg66280] Re: [mg66261] Re: Determining Linear dependent vectors
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Sat, 6 May 2006 23:50:47 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

On 6 May 2006, at 14:54, bghiggins at ucdavis.edu wrote:

> 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,
>

I wonder whether this is what the OP really meant by "determine  
linearly dependent vectors out of a given set"? Probably so, but just  
in case, here is a different interpretation of the question. Using  
the same example as above:

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


Minors[data, 4]


{{12}, {-16}, {0}, {48}, {24}}

This means that the set:


Drop[data, {3, 3}]


{{2, 0, 0, 1}, {1, 0, 1, 1}, {0, 0, 3, 0}, {0, 0, 0, 2}}

is a subset of 4 dependent vectors. Any other subset of the original  
set consisting of 4 or less vectors is independent.

Andrzej Kozlowski



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