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Coordinate vector basis proving question

  1. Mar 18, 2010 #1
    1. The problem statement, all variables and given/known data
    Let S = {v1,v2,...,vn} be a basis for an n-dimensional vector space V. Show that {[v1]s,[v2]s,...[vn]s} is a basis for Rn.
    Here [v]s means the coordinate vector with respect to the basis S.

    2. Relevant equations
    [v]s is the coordinate vector with respect to the basis S.


    3. The attempt at a solution
    S={v1..vn} is a basis and must be linearly independent.
    Any vector v in S then is a unique linear combination of the vectors in S, so v=a1v1+a2v2+...+anvn.
    Since [v]s in general = (a1,a2,...an), then every [vi]s where i = 1 .. n has a unique (a1,a2,...,an) and so the basis {[v1]s,...,[v2]s} will be linearly independent and thus form a basis for Rn.


    I have no answers to verify with, so I would like to know if I have answered it correctly. I am extremely weak with anything to do with proving so any assistance would be greatly appreciated, :).
     
  2. jcsd
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