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Math Help - Linear map and Linear independence

  1. #1
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    Linear map and Linear independence

    Hi! I've been trying to figure this out, but I'm not sure how I'd prove it..

    Let V, W be two vector spaces, and F: V-->W a linear map. Let w_1, ..., w_n be elements of W which are linearly independent, and let v_1, ..., v_n be elements of V such that F(v_i)=w_i for i = 1, ..., n. Show that v_1, ..., v_n are linearly independent.

    Help please! Thank you
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  2. #2
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    If it were a bijection then you could use the fact that F is linear, i.e.
     F(a_1v_1 + a_2v_2 + \cdots a_nv_n) = a_1F(v_1) + a_2F(v_2) + \cdots a_n F(v_n) = a_1w_1 + a_2w_2 + \cdots a_nw_n = 0 \Rightarrow a_1v_1 + a_2v_2 + \cdots a_nv_n = 0.
    Not sure if this is really that helpful though.
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  3. #3
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    You can't prove it, it's not true! A linear transformation maps independent vectors to independent vectors if and only if it is non-singular (invertible).

    A trivial counter-example is the linear transformation that maps any vector to the 0 vector.
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  4. #4
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    Yeah, that was my point HallsofIvy. The problem is that a linear transformation that maps more than one vector to the zero vector. For this to be true we must require the only vector mapped to the zero vector be the zero vector, otherwise the proof I wrote up doesn't work.
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