A set S of vectors is given. Find a subset of S with the same span as S that is as small as possible.
No idea how to do this. If you do your the best! Thanks
Solution
The smallest subset with the same span is { (1,-1,2) , (2,-3,0) }. To see this we take any representable vector in the span of the given set. a*(1,-1,2)+b*(2,-3,0)+c*(0,0,0). We may write the same arbitrary vector using linear combinations of { (1,-1,2) , (2,-3,0) }, ie. a*(1,-1,2)+b*(2,-3,0) = a*(1,-1,2)+b*(2,-3,0)+c*(0,0,0). Therefore the span is the same as any vector generated by the set is generated by the subset. We may show that the subset is the smallest such subset if it is linearly independant, ie. there is no constant c for which c*(1,-1,2) = (2,-3,0). If such a c were to exist then from the first component we have that c*1 = 2 therefore c=2, but in the third component we have c*2 = 0 there c=0, which shows no such c can satisfy both. Therefore the subset is as small as is possible whilst still having the same span as the given set.
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A set S of vectors is given- Find a subset of S with the same span as.docx
1. A set S of vectors is given. Find a subset of S with the same span as S that is as small as possible.
No idea how to do this. If you do your the best! Thanks
Solution
The smallest subset with the same span is { (1,-1,2) , (2,-3,0) }. To see this we take any
representable vector in the span of the given set. a*(1,-1,2)+b*(2,-3,0)+c*(0,0,0). We may write
the same arbitrary vector using linear combinations of { (1,-1,2) , (2,-3,0) }, ie. a*(1,-1,2)+b*(2,-
3,0) = a*(1,-1,2)+b*(2,-3,0)+c*(0,0,0). Therefore the span is the same as any vector generated by
the set is generated by the subset. We may show that the subset is the smallest such subset if it is
linearly independant, ie. there is no constant c for which c*(1,-1,2) = (2,-3,0). If such a c were to
exist then from the first component we have that c*1 = 2 therefore c=2, but in the third
component we have c*2 = 0 there c=0, which shows no such c can satisfy both. Therefore the
subset is as small as is possible whilst still having the same span as the given set.
