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Notation of linear operator

  1. Aug 7, 2008 #1

    Defennder

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    1. The problem statement, all variables and given/known data
    This is from a linear algebra textbook I'm reading. I don't know whether there is universal agreement as to how notation should read for this, so this thread may well be meaningless. But I'll just post it to see if anyone can decipher what the question means. I'm asked to prove it, by the way, but I can't do so unless I understand the notation:

    Theorem 6.11 Let V be an inner product space, and let T and U be linear operators on V. Then
    (b)[tex](cT)^* = \bar{c}T^* \ \mbox{for any c} \ \in F[/tex]


    2. Relevant equations
    3. The attempt at a solution
    I can understand the RHS of the question; that for a vector u in V, [tex]\bar{c}T^*(\vec{u})[/tex]. The * here denotes adjoint of the linear operator T, but how do I interpret the LHS? It clearly cannot be [tex]cT^*(\vec{u})[/tex]. Elsewhere, the book uses A* for a matrix A to denote the conjugate transpose of A, where every entry in A* is the complex transpose of the complex conjugate of the corresponding entry in A. But that can't be the interpretation either since if that is so, the LHS would be a row vector while the RHS a column vector.
     
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  3. Aug 7, 2008 #2

    CompuChip

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    For the left hand side: define a new linear operator [itex]S := c T[/itex]. Then the left hand side is [itex]S^*[/itex], so the adjoint of this new operator.

    The proof is easy by the way, just writing out a string of identities which follow from known (defining) properties.
     
  4. Aug 7, 2008 #3

    HallsofIvy

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    ?T is a linear operator. cT is the linear operator that maps vector v into c(T(v)). (cT)* is the adjoint of that operator. You say yourself "The * here denotes adjoint of the linear operator T". The fact that "Elsewhere, the book uses A* for a matrix A to denote the conjugate transpose of A, where every entry in A* is the complex transpose of the complex conjugate of the corresponding entry in A. " is not relevant here because you are talking about linear transformations, not matrices. Use the interpretation that is relevant.

    (Given a specific basis, a linear transformation can be written as a matrix. The two interpretations of "*" then coincide: the "adjoint of the linear transformation" has matrix (in that same basis) that is the "adjoint of the (original) matrix.)
     
  5. Aug 7, 2008 #4

    Defennder

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    I got it. The problem was that I didn't know how to interpret the notation. Thanks.
     
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