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Transition Matrix and Ordered Bases

  1. Apr 14, 2013 #1
    1. The problem statement, all variables and given/known data
    Let B and C be ordered bases for ℝn. Let P be the matrix whose columns are the vectors in B and let Q be the matrix whose columns are the vectors in C. Prove that the transition matrix from B to C equals Q-1P.


    2. Relevant equations
    An ordered basis for a vector space V is an ordered n-tuple of vectors (v1,...,vn) such that the set (v1,....,vn) is a basis for V.


    3. The attempt at a solution
    I know that if B is the standard basis in ℝn, then the transition matrix from B to C is given by [1st vector in C 2nd vector in C ........... nth vector in C]-1.

    Also, if C is a standard basis in ℝn, then the transition matrix from B to C is given by [1st vector in B 2 vector in B ......... nth vector in B].

    Since I konw what the transition matrix is from B to C given different standard bases, I am having a difficult time relating this to teh columns of each.





    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
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