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Math Help - Proof for Isomorphic Linear Operator

  1. #1
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    Proof for Isomorphic Linear Operator

    The following linear algebra problem on linear transformations/isomorphisms was confusing me so any help would be appreciated!

    Let R^n -> R^n be a linear operator with matrix A (using the standard basis for R^n). Prove that L is an isomorphism if and only if the columns of A are linearly independent.
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  2. #2
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    Quote Originally Posted by buckaroobill View Post
    The following linear algebra problem on linear transformations/isomorphisms was confusing me so any help would be appreciated!

    Let R^n -> R^n be a linear operator with matrix A (using the standard basis for R^n). Prove that L is an isomorphism if and only if the columns of A are linearly independent.
    Let L be the linear operator.

    "Isomorphism --> Linear independence".
    Isomorphism (for vector spaces) means a bijective linear operator. Bijective means both injective and surjective. Let b in R^n then there exists a unique x in R^n such that L(x)=b. That is, Ax = b. (Why?).
    Thus, the linear system of equations,
    Ax = b has a unique solution for every column vector b.
    Thus, det(A) != 0.
    Thus, the column and row vectors in A are linearly independent.
    Q.E.D.

    See if you can do the converse.
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