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Linear algebra, when does the implications hold?

  1. Dec 7, 2013 #1
    Hi, I have 4 implications I am interested in, I think I know the answer to the first 2, but the last two is not something I know, however they are related to the first 2 so I will include all to be sure.

    Assume that T is a linear transformation from from vectorspace A to B.

    T: A -> B
    A* is n vectors in A, that is A* = {a1, a2, an}

    1.
    T(A*) linearly independent -> A* linearly independent
    If T(A*) is linearly independent, then A* must be linearly independent, without any requirements for T?

    2.
    T(A*) lindearly dependent -> A* linearly dependent
    If T(A*) is linearly dependent, then we can only conclude that A* is linearly independent only if T is 1-1, if T is not 1-1 we can not conlude anything?

    3.
    span(T(A*))=B -> span(A*)= A
    If span(T(A*)) = B, what requirement must we have to conlude that span(A*) = A. That T is 1-1, surejective, both or none?

    4.
    span(T(A*)) ≠ B -> span(A*) ≠ A*
    If span(T(A*)) is not B what must we have to conlude that span(A*) is not A? Will this implication hold if T is 1-1, surjective, both or none?

    thanks
     
    Last edited: Dec 8, 2013
  2. jcsd
  3. Dec 8, 2013 #2
    1. If c1a1+...+cnan=0, then T(c1a1+...+cnan)=0. So no restrictions.
    2. Injectivity is not needed. Consider T: R^3->R^2. T(x,y,z)=(x,y). Surjectivity is not needed. Consider
    T: R^2->R^3. T(x,y)=(x,y,0).
    3. Let B={0} and you'll see surjectivity isn't enough. 1-1 is. Think of any x in A and consider T(x)=c1T(a1)+c2T(a2)+...+cnT(an).
    4. Your last statement is equivalent to span(A*)=A -> span(T(A*)) = B. If dim(A)<dim(B) and you have n=dim(A) vectors, this is not true. So injectivity is not enough. Surjectivity is though. Consider y in B. Then there is an x in A such that T(x)=y. Or T(c1a1+...cnan)=y.
     
  4. Dec 8, 2013 #3

    SteamKing

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    The word is 'linearly'.
     
  5. Dec 8, 2013 #4

    Hey, thanks man.
    I just have some more questions, are you allowed to prove a general statement with examples?

    Also, i am not sure about the example.
    T(x,y,z) = (x,y)
    if A* = {(1,1,0),(1,1,1)}
    then T(A*) = {(1,1),(1,1)}
    so T(A*) is linearly dependent, but A* is not, so the implication is false here.
     
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