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Linear Algebra help-invertibility

  • 30-04-2011 9:26am
    #1
    .E_C_K_S.
    Registered Users, Registered Users 2 Posts: 1,150 ✭✭✭


    Hi lads. I'm just wondering can anyone help me out with this as I don't know where to start...Here's the question

    Suppose that T:V->V is a linear map which satisfies T^5=0

    (i) Prove that T is not invertible
    (ii)Prove that either T=0 or else T is not diagonalizable.


    Another similar question:

    Suppose that T:V->V is a linear map on a finite dimensional vector space. Suppose that (T-I)^1000=0

    (i) Prove that T is invertible
    (ii) Prove that either T=I or else T is not diagonalizable

    Thanks!


Comments

  • #2
    Eliot Rosewater
    Registered Users, Registered Users 2 Posts: 3,745 ✭✭✭Eliot Rosewater


    First question part (i) - you've it down wrong there, should be "Prove that T is not invertible." Anyway, assume it is, see what happens.

    Two facts to help with the rest:
    • A linear transformation is invertible if and only if 0 is not an eigenvalue.
    • A linear transformation is diagonalisable if and only if it has n linearly independent eigenvectors (n the dimension of the vector space).


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