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Homework Help: Quick question

  1. Oct 16, 2005 #1
    Do cancelling rules work for vectors? For example, if you had
    [tex]A\vec{x} = B\vec{x}[/tex]
    could you cancel out the x's and be left with A = B? Is that legal?

    edit: A and B are matrices. Does this make a difference?
     
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  3. Oct 16, 2005 #2

    Tom Mattson

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    Let's fill in some missing steps and then the answer to this question will become clearer.

    [tex]A\vec{x}=B\vec{x}[/tex]
    [tex]A^{-1}A\vec{x}=A^{-1}B\vec{x}[/tex]
    [tex]I\vec{x}=I\vec{x}[/tex]
    [tex]\vec{x}=\vec{x}[/tex]

    Now let me ask you: What did I have to assume in order to write each line?
     
  4. Oct 16, 2005 #3
    So the assumption was that the matrix A = the matrix B. So then we could take the inverse of each to form the identity matrix on each, and then it followed that Ix = Ix so x=x. So then if we assume that matrices are basically just equivalents of multi-column vectors, and we can cancel them out from both sides, then we should be able to do the same for vectors.

    Is that right?
     
  5. Oct 16, 2005 #4

    Tom Mattson

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    You're right that I had to assume that [itex]A=B[/itex] in order to get [itex]A^{-1}B=I[/itex].

    No, there's more to it than that. In the second line I had to assume that [itex]A[/itex] is invertible. And as you would have learned from class, nonsquare matrices are not invertible. So the matrices [itex]A[/itex] and [itex]B[/itex] can't just be any collection of column (or row) vectors.

    Note that the case that [itex]A[/itex] is not invertible includes the case that [itex]A=0[/itex] (the zero matrix). In that case the equation [itex]A\vec{x}=B\vec{y}[/itex] is satisfied if [itex]\vec{y}=\vec{0}[/itex] (the zero vector) and [itex]B[/itex] is any matrix of appropriate dimensions. Note that [itex]\vec{x}[/itex] is not necessarily equal to [itex]\vec{y}[/itex] in this case.
     
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