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Linear Transformations and Basis

  1. Oct 18, 2011 #1
    1. The problem statement, all variables and given/known data

    Show that if [tex]{ v_1, ... , v_k} [/tex] spans [tex] V [/tex] then [tex] {T(v_1), ... , T(v_k)}[/tex] spans [tex] T(v) [/tex]


    2. Relevant equations



    3. The attempt at a solution

    So we know that every vector in V can be written as a linear combination of [tex] v_1,...v_k [/tex] thus we only need to show that [tex] {T(v_1), ... , T(v_k)}[/tex] spans [tex] T(c_1v_1 + ... + c_kv_k) [/tex]

    However, I'm not sure how to do that.
     
  2. jcsd
  3. Oct 18, 2011 #2

    micromass

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    You probably meant to write T(V) here...

    What in earth does that even mean??????

    You need to show that [itex]\{T(v_1),...,T(v_k)\}[/itex] spans T(V). So take a vector in T(V) and show that it can be written as

    [tex]T(c_1v_1 + ... + c_kv_k) [/tex]
     
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