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Math Help - Linear transformations and linear independance

  1. #1
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    Linear transformations and linear independance

    Hi,

    I was curious as to whether this statement was true or not, If it's true then can anyone show me a hint for the proof of it.

    "If we have a linear transformation f:V -> V and if v(1),v(2),...,v(n) are linearly independant vectors in V, then does that imply that f(v(1)),f(v(2)),...f(v(n)) are also linearly independent in V. We are also told that f is injective, so f(v(k)) cannot equal 0, if v(k) is not equal to 0, since we know that with linear transformations f(0)=0."

    v(i) means: v subscript i, for all i in [1,2,...,n],
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  2. #2
    Junior Member Nehushtan's Avatar
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    Re: Linear transformations and linear independance

    Suppose a_1f(v_1)+\cdots+a_nf(v_n)=0. As f is a linear transformation, a_1f(v_1)+\cdots+a_nf(v_n)=f(a_1v_1+\cdots+a_nv_n). Therefore f(a_1v_1+\cdots+a_nv_n)=0=f(0). As f is injective, this implies a_1v_1+\cdots+a_nv_n=0, and the linear independence of v_1,\ldots,v_n implies a_1=\cdots=a_n=0
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  3. #3
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    Re: Linear transformations and linear independance

    Of course, without the condition that the linear transformation is injective, this would be no longer true.
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