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Homework Help: Linear algebra - matrices and polynomials

  1. Jan 17, 2010 #1
    1. The problem statement, all variables and given/known data
    Prove for each square matrix B there is a real polynomial p(x) (not the zero polynomial) so p(B)=0


    2. Relevant equations
    Rank-nullity? dimv = r(T) + n(T)


    3. The attempt at a solution
    I've found the dimension for nxn square matrices (n²) and a basis (1 in one place and zero in the rest for each place in the matrix) but I don't really know where to go from here. Any ideas would be great :) thanks.
     
  2. jcsd
  3. Jan 17, 2010 #2
    Consider,
    [tex]1,B,B^2,B^3,...,B^{n^2}[/tex]
    is this set linearly dependent? (can n^2+1 many vectors be linearly independent in a n^2 dimensional vector space?) If they are linearly dependent you can find coefficients a0,a1,a2,...,a(n^2) such that,
    [tex]\sum_{i=0}^{n^2} a_i B^i = 0[/tex]
    so consider,
    [tex]p(x) = \sum_{i=0}^{n^2} a_i x^i[/tex]
     
  4. Jan 17, 2010 #3
    Thank you, that helped a lot.
     
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