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Homework Help: The relation between span(In,A,A2, )and it's minimal polynomial

  1. Jan 27, 2013 #1
    Let A ∈ Mn×n(F )



    Why dim span(In, A, A2, A3, . . .) = deg(mA)?? where mA is the minimal polynomial of A.



    For span (In,A,A2...)

    I can prove its

    dimension <= n by CH Theorem

    but what's the relation between

    dim span(In,A,A2...)and deg(mA)
     
  2. jcsd
  3. Jan 27, 2013 #2
    For example, if the minimal polynomial is [itex]x^2+x+1[/itex]. Then [itex]A^2+A+1=0[/itex].

    Do you see any way to conclude that [itex]A^2\in span(A,1)[/itex]?
     
  4. Jan 27, 2013 #3
    ybut i dont know why cant I write A
     
  5. Jan 27, 2013 #4
    ybut i dont know why cant I write A in spanA2I
     
  6. Jan 27, 2013 #5
    y,but i dont know why cant I write A in span{A2,I}?
    I'm sorry about the type..
     
  7. Jan 27, 2013 #6
    That's also true, but I don't see how this fact helps you solve the problem.
     
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