1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    User Avatar
    Science Advisor
    Homework Helper

    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


    User Avatar
    Science Advisor
    Homework Helper

    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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Linear independance
  1. Linear Independence (Replies: 1)

  2. Linear independence (Replies: 2)

  3. Linear independence (Replies: 6)

  4. Linear independance (Replies: 3)

  5. Linear independence (Replies: 3)