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Definition of Image of a linear transformation

  1. Feb 3, 2016 #1
    1. The problem statement, all variables and given/known data
    The image of a linear transformation = columnspace of the matrix associated to the linear transformation.
    More specifically though, given the transformation from Rn to Rm: from subspace X to subspace Y, the image of a linear transformation is equal to the set of vectors in X that are mapped to Y. This may or may not be equal to the all of the vectors in subspace X and subspace Y.

    I was going to say, the Im(T) = all of the vectors in X that are mapped to Y, but the definition sounds a bit 'muddier', but I'm not entirely sure. Hence my post.

    I usually draw a picture like this: http://thejuniverse.org/PUBLIC/LinearAlgebra/MATH-232/Unit.8/Presentation.1/Section7B/image.png to go with my definition, but I wanted to check.

    I saw a reference book that said 'the image of a linear transformation f : V → W is the set of vectors the linear transformation maps to.'



    2. Relevant equations


    3. The attempt at a solution
     
  2. jcsd
  3. Feb 3, 2016 #2

    blue_leaf77

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    The image of a linear transformation ##T## defined by ##T:V\rightarrow W## is the set of all vectors which are the result of the linear map ##T## applied on any vector in ##V##. Which means ##\textrm{Im}(T)## is a subset of ##W##, not of ##V##. In this sense, the last definition:
    is the correct one.
     
  4. Feb 3, 2016 #3

    HallsofIvy

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    No! The "image of a linear transformation" is set of vectors in Y such that some vector in X is mapped to it. Taking the linear transformation to be "T", the image is "all y in Y such that there exist x in X such that Tx= y."

    The image is a subset (actually subspace) of Y, not X. It may or may not be all of Y.

    Again, no, no, no! Im(T) = all vectors in Y that vectors in X are mapped to it.


    The pink area in your picture is the image.

    Yes, that is correct, NOT what you said!



     
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