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Math Help - Linearly independent set

  1. #1
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    Linearly independent set

    If A is a 3x3 matrix and {v1, v2, v3} is a linearly independent set of vectors in R^3 for which {Av1, Av2, Av3} is also a linearly independent set, then the matrix A must be invertible.

    Is this true? Can someone please explain why or why not??

    I have read a lot but could not find any resources related to this theorem.
    I do not have any idea of how to look at this problem.
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  2. #2
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    Quote Originally Posted by mola View Post
    If A is a 3x3 matrix and {v1, v2, v3} is a linearly independent set of vectors in R^3 for which {Av1, Av2, Av3} is also a linearly independent set, then the matrix A must be invertible.

    Is this true? Can someone please explain why or why not??

    I have read a lot but could not find any resources related to this theorem.
    I do not have any idea of how to look at this problem.

    If the image of a basis is again a basis the transformation (or matrix) must be an isomophism (invertible), since its kernel is zero. Check it.
    Of course, the above applies for finite dimensions...

    Tonio
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