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Math Help - Matrix and endomorfism

  1. #1
    Junior Member
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    Exclamation Matrix and endomorfism

    Hello

    Could you please help me with this problem?

    Prove that a matrix associated with an endomorfism T has inverse if and only if T is injective.

    Thanks a lotĦĦ
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by osodud View Post
    Hello

    Could you please help me with this problem?

    Prove that a matrix associated with an endomorfism T has inverse if and only if T is injective.

    Thanks a lotĦĦ
    Well, a matrix T has an inverse if and only if...what? There are so many properties that finish off this sentence, but pick your favourite. Does this property tell you anything about the matrix begin injective? Surjective?

    For instance, if you go for the obvious property, T is invertible if and only if there exists a matrix S such that ST=I=TS. Then, suppose T is not injective. that means there exists some non-zero vector which is mapped to zero (that is to say, ker(T) \neq \{0\}). Call this vector v. Then vI=v but vTS = 0S=0 an contradiction. Thus the matrix must be injective.

    Now, suppose that your matrix is an injective endomorphism and prove that this implies that it is invertible.
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