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