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Math Help - Linear transformation question

  1. #1
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    Linear transformation question

    I've got a question like this:

    Let V be a finite dimensional vector space with dimV >= 2 . Show
    that there exists a linear transformation T:V to V such that T not equal to zero but
    T^2 = 0 .

    I have figured out the "T not equal to zero but T^2 = 0 ." part already, rankT>=1, dimNulT>=rankT>=1, satisfies with dimV>=2 by rank theorem, but how to present it nicely?
    Last edited by wesleybrown; October 28th 2012 at 06:51 AM.
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  2. #2
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    Re: Linear transformation question

    pick a basis B = {b1,...,bn} for V.

    thus we can write any v in V as v = c1b1+...+cnbn.

    define T(c1b1+...+cnbn) = cnb1.

    T is non-zero because T(bn) = b1 ≠ 0, since b1 is a basis vector.

    but T2(v) = T(T(v)) = T(T(c1b1+...+cnbn)) = T(cnb1) = 0

    (because cnb1 = cnb1 + 0b2+....+0bn).

    basically we are using the linear transformation that has the matrix (relative to the basis B):

    \begin{bmatrix}0&0&\dots&0&1\\0&0&\dots&0&0\\ \vdots&\vdots&\ddots&\vdots&\vdots\\0&0&\dots&0&0 \end{bmatrix}
    Thanks from wesleybrown
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