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I Does this theorem need that Ker{F}=0?

  1. May 1, 2017 #1
    I have encountered this theorem in Serge Lang's linear algebra:
    Theorem 3.1. Let F: V --> W be a linear map whose kernel is {O}, then If v1 , ... ,vn are linearly independent elements of V, then F(v1), ... ,F(vn) are linearly independent elements of W.

    In the proof he starts with C1F(v1) + C2F(v2) + ... + CnF(vn) and then uses linearity and injectivity to prove that the constants are 0.

    I can't see where injectivity is essential, he could have proved it with linearity alone.
    He arrives at a point where we have :

    F(C1V1 + ... + CnVn) = 0

    He uses injectivity here to prove that C1V1 + ... + CnVn = 0 and since v1 ... vn linearly independent, the constants are 0. But that can be also solved by the fact that F is linear.
     
  2. jcsd
  3. May 1, 2017 #2

    martinbn

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    Science Advisor

    Say ##F(v_1)=0##, then ##1*F(v_1)+0*F(v_2)+\cdots+0*F(v_n)=0##, they are dependent.
     
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