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Math Help - Prove linear transformation is invertible

  1. #1
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    Prove linear transformation is invertible

    I need to prove that if T is a linear transformation from R^n to R^n, which in onto, then T is invertible.

    I am not sure where to begin. Any help would be great.
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  2. #2
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    Re: Prove linear transformation is invertible

    Quote Originally Posted by page929 View Post
    I need to prove that if T is a linear transformation from R^n to R^n, which in onto, then T is invertible.

    I am not sure where to begin. Any help would be great.

    If T:V_1\to V_2 is a linear transformation then:

    1. it is invertible \iff it is onto and 1-1,

    2. it is 1-1 \iff\, Ker(f) = \{O\},

    3. it is onto \iff Im(f) = V_2,

    4. \text{dim}\, V_1 = \text{dim} \,Ker(f) + \text{dim}\, Im(f).
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  3. #3
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    Re: Prove linear transformation is invertible

    Are you allowed to use the "rank-nullity" theorem, that if A:U->V then the dimension of AU plus the dimension of the kernel of A is equal to the dimension of U? If A is "onto" R^n, its rank is n. Since the A is from R^n, its nullity is n- n= 0 so A is also "one to one".
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