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Math Help - True or false linear transformation

  1. #1
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    True or false linear transformation

    Are the following true or false? If it is true, prove it, if it is false give a counter example (and proof that your counter example is true).

    1) If T is a linear transformation such that T: R^2 -> R^3, T can't be one to one.

    2) If a matrix has characteristic polynomial f(x) = (x-2)^2(x+4)(x-4), then the matrx is not diagonalizable.
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    Senior Member bkarpuz's Avatar
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    Quote Originally Posted by cindy View Post
    Are the following true or false? If it is true, prove it, if it is false give a counter example (and proof that your counter example is true).

    1) If T is a linear transformation such that T: R^2 -> R^3, T can't be one to one.
    T(x,y)=(x,y,0) is one to one.
    May be you should ask T:\mathbb{R}^{3}\to\mathbb{R}^{2}?
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  3. #3
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    Quote Originally Posted by cindy View Post
    1) If T is a linear transformation such that T: R^2 -> R^3, T can't be one to one.
    Like bkarpuz said it should be the other way around if you want to have an impossibility.
    In general T: \mathbb{R}^n \to \mathbb{R}^m is not one-to-one if n>m by rank-nullity theorem.

    2) If a matrix has characteristic polynomial f(x) = (x-2)^2(x+4)(x-4), then the matrx is not diagonalizable.
    \begin{bmatrix} 2&0&0&0\\0&2&0&0\\0&0&-4&0\\0&0&0&4 \end{bmatrix}
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