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Math Help - Linear Algebra Proof

  1. #1
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    Linear Algebra Proof

    Let u and v be vectors in Rn, and let T be a linear operator on Rn. Prove that T(n) * T(v) = n * v if and only if AT = A-1 where A is the standard matrix for T.

    Any help would be great.
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  2. #2
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    Re: Linear Algebra Proof

    What does n mean? Does it mean a vector normal to v?

    The transpose equaling the inverse means that the columns of A form an orthonormal basis.
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  3. #3
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    Re: Linear Algebra Proof

    i think it's a typo. i believe the OP is asking to show that every inner-product preserving linear operator has an orthogonal matrix in the standard orthonormal basis, and that an orthogonal matrix represents an inner-product preserving linear operator.

    the whole of this proof rests on the fact that (for the standard inner product in Rn):

    \mathbf{u}\cdot \mathbf{v} = \mathbf{v}^T\mathbf{u}

    so if:

    T\mathbf{u} \cdot T\mathbf{v} = \mathbf{u}\cdot \mathbf{v}

    for all u, v, then:

    ((A\mathbf{v})^T)(A\mathbf{u}) = \mathbf{v}^T\mathbf{u}

    that is:

    \mathbf{v}^TA^TA\mathbf{u} = \mathbf{v}^T\mathbf{u}

    for all u, so

    \mathbf{v}^TA^TA = \mathbf{v}^T

    so that:

    (\mathbf{v}^TA^TA)^T = (\mathbf{v}^T)^T

    A^T(A^T)^T\mathbf{v} = \mathbf{v} = I\mathbf{v}

    for all v, thus:

    A^TA = I

    A^T = A^{-1}

    these steps are all reversible.
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