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Math Help - Linear transformation relative to a basis:A question

  1. #1
    Senior Member x3bnm's Avatar
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    Linear transformation relative to a basis:A question

    My problem:
    Suppose
    1)Matrix for linear transformation T relative to basis B \text{ is}: A
    2)Matrix for linear transformation T relative to basis B^\prime \text{ is}: A^\prime
    3)Transition matrix from basis B^\prime to basis B is: P
    4)Transition matrix from basis B to basis B^\prime is : P^{-1}

    Now we know that:
     A^\prime [\mathbf{v}]_{B^\prime} = [T(\mathbf{v})]_{B^\prime}

    And
      P^{-1}AP[\mathbf{v}]_{B^\prime} = [T(\mathbf{v})]_{B^\prime}
    where [T(\mathbf{v})]_{B^\prime} is coordinate matrix of T(\mathbf{v}) relative to basis B^\prime and  [\mathbf{v}]_{B^\prime} is coordinate matrix of \mathbf{v} relative to B^\prime

    Now the book(Elementary Linear Algebra by Larson 6th edition) I am reading states that:

    "But by the definition of the matrix of a linear transformation relative to a basis, this implies that:
      A^\prime = P^{-1}AP
    "

    My question:
    My question is: why's that?

    I know from above that:
      [T(\mathbf{v})]_{B^\prime} = A^\prime [\mathbf{v}]_{B^\prime} = P^{-1}AP[\mathbf{v}]_{B^\prime}

    which makes:
      A^\prime [\mathbf{v}]_{B^\prime} = P^{-1}AP[\mathbf{v}]_{B^\prime}

    But what reasoning makes the statement A^\prime = P^{-1}AP true? How did the author deduced this to be true?
    Last edited by x3bnm; November 4th 2011 at 09:22 AM.
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  2. #2
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    Re: Linear transformation relative to a basis:A question

    Quote Originally Posted by x3bnm View Post
    My problem:
    Suppose
    1)Matrix for linear transformation T relative to basis B \text{ is}: A
    2)Matrix for linear transformation T relative to basis B^\prime \text{ is}: A^\prime
    3)Transition matrix from basis B^\prime to basis B is: P
    4)Transition matrix from basis B to basis B^\prime is : P^{-1}

    Now we know that:
     A^\prime [\mathbf{v}]_{B^\prime} = [T(\mathbf{v})]_{B^\prime}

    And
      P^{-1}AP[\mathbf{v}]_{B^\prime} = [T(\mathbf{v})]_{B^\prime}
    where [T(\mathbf{v})]_{B^\prime} is coordinate matrix of T(\mathbf{v}) relative to basis B^\prime and  [\mathbf{v}]_{B^\prime} is coordinate matrix of \mathbf{v} relative to B^\prime

    Now the book(Elementary Linear Algebra by Larson 6th edition) I am reading states that:

    "But by the definition of the matrix of a linear transformation relative to a basis, this implies that:
      A^\prime = P^{-1}AP
    "

    My question:
    My question is: why's that?

    I know from above that:
      [T(\mathbf{v})]_{B^\prime} = A^\prime [\mathbf{v}]_{B^\prime} = P^{-1}AP[\mathbf{v}]_{B^\prime}

    which makes:
      A^\prime [\mathbf{v}]_{B^\prime} = P^{-1}AP[\mathbf{v}]_{B^\prime}

    But what reasoning makes the statement A^\prime = P^{-1}AP true? How did the author deduced this to be true?
    if two functions are the same for every input of the domain, they are the same function.

    so if A' and P^{-1}AP are the same for every [\mathbf{v}]_{B'}, they are the same function (and if they are both linear transformations, the same linear transformation).
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  3. #3
    Senior Member x3bnm's Avatar
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    Re: Linear transformation relative to a basis:A question

    Quote Originally Posted by Deveno View Post
    if two functions are the same for every input of the domain, they are the same function.

    so if A' and P^{-1}AP are the same for every [\mathbf{v}]_{B'}, they are the same function (and if they are both linear transformations, the same linear transformation).
    Thanks for reply. But suppose three matrices and [x] are such that:
    ABx = CBx where AB = CB for all [x]

    Then by the properties of matrix:
    A is not necessarily equal to C

    So for A^\prime [\mathbf{v}]_{B^\prime} = P^{-1}AP[\mathbf{v}]_{B^\prime}

    why my reasoning above is not a problem for this situation?

    Why still   A^\prime = P^{-1}AP  ?
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  4. #4
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    Re: Linear transformation relative to a basis:A question

    there's no cancellation going on, we're saying A' and P^-1AP are the same function (the fact that they have the same matrix relative to the basis B' is a big clue).
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  5. #5
    Senior Member x3bnm's Avatar
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    Re: Linear transformation relative to a basis:A question

    Quote Originally Posted by Deveno View Post
    there's no cancellation going on, we're saying A' and P^-1AP are the same function (the fact that they have the same matrix relative to the basis B' is a big clue).
    Thanks a lot. That I understand.
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