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Math Help - Linear Algebra Linear maps dealing with linear independence.

  1. #1
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    Linear Algebra Linear maps dealing with linear independence.

    Hi, I have this proof but I just want to know if I'm doing this right, or if there's a better way to write my proof (which is incomplete).

    5. Let V and W be vector spaces over F, and suppose that T in L(V,W) is injective. Given a linearly independent list (v1, ..., vn) of vectors in V, prove that the list (T(v1), ..., T(vn)) is linearly independent in W.

    So, I think the idea behind proving this is that, injective means that every element of dom(T) has a unique mapping to range(T), thus when you have a list of vectors of the mapping of every vector in the (v1, ... vn), each mapping will be unique, and thus that list in linearly independent?

    I'm a bit all over the place, and I guess i'm just having trouble finding a way to put this, and also wondering if I'm missing anything important.
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    Re: Linear Algebra Linear maps dealing with linear independence.

    Adding more thought, what exactly would imply that (w1, ..., wn), from T(v1, ..., vn)=(w1, ..., wn) for a vector in W, is linearly independent too from the fact that (v1, ..., vn) is linear independent and T is injective?
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  3. #3
    MHF Contributor Siron's Avatar
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    Re: Linear Algebra Linear maps dealing with linear independence.

    I would prove it by contraposition.
    Suppose \{T(v_1),\ldots, T(v_n)\} is linearly dependent then we could for example write T(v_1) as a linear combination of the other vectors, say
    T(v_1) = \sum_{i=2}^{n} k_i T(v_i) with \forall i \in \{2,\ldots,n\}: k_i \in \mathbb{F}. Since T is a linear map we can write T(v_1) = T\left(\sum_{i=2}^{n} k_i v_i\right), as T is injective we have v_1 =  \sum_{i=2}^{n} k_i v_i but that means \{v_1,\ldots,v_n\} is linearly dependent.
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    Re: Linear Algebra Linear maps dealing with linear independence.

    Thank you but the proof was to prove that {T(v1), ..., T(vn)} was linearly Independent.
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  5. #5
    MHF Contributor Siron's Avatar
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    Re: Linear Algebra Linear maps dealing with linear independence.

    Quote Originally Posted by zachoon View Post
    Thank you but the proof was to prove that {T(v1), ..., T(vn)} was linearly Independent.
    Yes, that's the reason I said I would give a proof by contraposition.
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