Linear algebra - matrices and polynomials

  • Thread starter Kate2010
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  • #1
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Homework Statement


Prove for each square matrix B there is a real polynomial p(x) (not the zero polynomial) so p(B)=0


Homework Equations


Rank-nullity? dimv = r(T) + n(T)


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.
 

Answers and Replies

  • #2
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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]
 
  • #3
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Thank you, that helped a lot.
 

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