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Math Help - Linear transformation and indepence problem

  1. #1
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    Linear transformation and indepence problem

    I am having trouble with a problem and am in need of some help.
    V & W are vector space, and T:V->W is linear.
    T is 1 to 1 and S is a subset of V. Prove that S is linearly independent if and only if T(S) is linearly independent.

    I have started the problem but am really uncertain of my approach. So far I have
    S=(a1,a2,...,an)
    V=(a1,a2,...,an,an+1,...,an+k)

    Sum [biT(ai)] = 0 where i=1,2,...n
    T[Sum bi(ai)] = 0

    therefore there should exist some c such that

    Sum [bi(ai)] = Sum ci(ai)

    Is what I have done even correct?
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  2. #2
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    Quote Originally Posted by lllll View Post
    I am having trouble with a problem and am in need of some help.
    V & W are vector space, and T:V->W is linear.
    T is 1 to 1 and S is a subset of V. Prove that S is linearly independent if and only if T(S) is linearly independent.
    Let S = \{ a_1, ... a_n\} and suppose this set is linearly independent. Then T(S) = \{T(a_1),...,T(a_n)\}. Suppose there exists c_1,...,c_n (in the field) such that c_1T(a_1) + ... + c_nT(a_n) = 0 thus T(c_1a_1+...+c_na_n) = 0. But since T is one-to-one its kernel is the zero-vector thus c_1a_1+...+c_na_n \implies c_1 = c_2 = ... = c_n. Thus, \{T(a_1),...,T(a_n) \} are linearly independent.
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  3. #3
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    Would it be a similar prove if S was a basis instead of a subset?
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