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Homework Help: Linear independance

  1. Feb 21, 2006 #1
    Suppose that {v1,v2,...,vn} is a minimal spanning set for a vector space V. That is V = span {v1,v2,...,vn} and V cannot be spanned by fewer than n vectors. Show that {v1,v2,...,vn} is a basis of V

    to be a basis for V then V = span (v1,v2,...,vn}
    we already have that
    suppose one of thise vectors was not linearly dependant then the number of vectors in the span is less than n. But V = span of n vectors
    so the set of vectors must be lienarly independant
  2. jcsd
  3. Feb 22, 2006 #2

    matt grime

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    Surely the definition of a basis is that it is a minimal spanning set?

    Or are you starting from linearly independent spanning set and showing any such thing is minimal, and vice versa?
  4. Feb 22, 2006 #3


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    Yes, to be minimal, they have to be independent. Suppose not, then you could do away with at least one (how?) and still span V (you should work out an example here), which contradicts "minimal."

    See http://en.wikipedia.org/wiki/Vector_basis
    Last edited: Feb 22, 2006
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