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Homework Help: I cant understand some terminology

  1. Feb 8, 2008 #1
    "T(1,2)=(3,4)
    T(T(1,2))=T(3,4)=(3,4)

    (1,2) and (3,4) are linearly independant and form a basis of R2, so (3,4) spans ImT. Therefore dim(imT)=1, and obviously dim(kerT)=1, so the transformation is singular."

    this is a solution to some problem
    i cant understand what is "spans an image"
    i got
    T(1,2)=(3,4)
    and
    T(3,4)=(3,4)

    i know that the image is the resolt of putting the vector in the transformation

    but what meens

    "(3,4) spans ImT"
    ??
     
  2. jcsd
  3. Feb 8, 2008 #2

    arildno

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    That it spans the image of T means that any element in ImT can be represented as [itex]\gamma*(3,4)[/itex] for some real number [itex]\gamma[/itex]

    Now, to prove this, an ARBITRARY vector in R2 can be represented as:
    [tex]\vec{v}=\alpha(3,4)+\beta(1,2)[/itex] since those two are linearly independent.

    Therefore using the properties of a linear transformation, we get:
    [tex]T(\vec{v})=\alpha{T}(3,4)+\beta{T}(1,2)=\gamma(3,4), \gamma=\alpha+\beta[/tex]
     
  4. Feb 8, 2008 #3
    thanks
     
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