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Proof involving linear transformation of a set of vectors

  1. Sep 29, 2010 #1
    1. The problem statement, all variables and given/known data
    Let T:[tex]\Re^{n}[/tex][tex]\rightarrow[/tex][tex]\Re^{m}[/tex] and let S={u,v,w}[tex]\in[/tex][tex]\Re^{n}[/tex].

    If S is linearly dependent, show that {T(u), T(v), T(w)} is also linearly dependent.

    2. Relevant equations

    3. The attempt at a solution
    Since S[tex]\in[/tex][tex]\Re^{n}[/tex] then S`[tex]\in[/tex][tex]\Re^{m}[/tex].
    Not sure where to go from here
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Sep 29, 2010 #2
    If the set of vectors are dependent you can find a linear relationship between them.

    What happens to this linear relationship under the map T?
  4. Sep 29, 2010 #3
    If p(t) is the linear relationship between the vectors in S, does that mean that T[p(t)] is the linear relationship between the vectors in S`?

    If so I guess that proves that S` is also linearly dependent since there exists the linear relationship T[p(t)] between its elements.
  5. Sep 30, 2010 #4
    Look up the definition of linear dependence.
  6. Sep 30, 2010 #5
    Something is lacking in your statement of the problem. Is T supposed to be a linear map? You did not say so.
  7. Sep 30, 2010 #6


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    Don't ask, work it out. Your "p(t) is the linear relationship" is what Ziox suggested you look up- the definition of "linearly dependent". If vectors, u, v, and w are linearly dependent, then there exist numbers, a, b, c, not all 0, such that au+ bv+ cw= 0.

    Now look at T(au+ bv+ cw)= T(0).

    As Arkajad suggested, you will need to specify that T is a linear transformation (which you do in a later post).
  8. Sep 30, 2010 #7
    Aha! I think I have it.

    Since T is a linear transformation, T(0)=0, and T(au+ bv+ cw)=aT(u)+bT(v)+cT(w)=0

    Since there exists scalars, a, b, c, not all 0, such that the above equation is true, S` is linearly dependent.

    I cant believe I missed that even after ZioX's advice, guess I needed a study break. Thanks guys.
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