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Math Help - Linear Independence and Transformation Proof Help

  1. #1
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    Linear Independence and Transformation Proof Help

    Does anyone know how to prove the following:

    The set of images of a linearly independent set of vectors under a linear transformation are also linearly independent?

    Thanks.
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  2. #2
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    Re: Linear Independence and Transformation Proof Help

    This is true only for injective transformations. For example, the projection of two-dimensional vectors on the horizontal axis maps any two vectors into linearly dependent vectors.
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    Re: Linear Independence and Transformation Proof Help

    What would be the best way to prove this?
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  4. #4
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    Re: Linear Independence and Transformation Proof Help

    Prove what? The original claim is false in general.
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    Re: Linear Independence and Transformation Proof Help

    What is the best way to answer part 2?

    Prove that the set of images of a linearly dependent set of vectors under a linear transformation is linearly dependent. Let {v1, , , vp} be a set of linearly dependent vectors, and let T: Rn Rm be a linear transformation. Show that {T(v1), , T(vp)} is linearly dependent. Part 2: Is it also true that the images of a linearly independent set of vectors under a linear transformation is also linearly independent? Explain.

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  6. #6
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    Re: Linear Independence and Transformation Proof Help

    Quote Originally Posted by PiIsCool View Post
    What is the best way to answer part 2?

    Part 2: Is it also true that the images of a linearly independent set of vectors under a linear transformation is also linearly independent? Explain.
    I answered this in post #2.
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  7. #7
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    Re: Linear Independence and Transformation Proof Help

    suppose there exist c1,c2,...,cn not all 0 with:

    c1v1 + c2v2 +...+ cnvn = 0

    (that is {v1,v2,...,vn} is a linearly dependent set), and that T is a linear transformation.

    then T(c1v1 + c2v2 +...+ cnvn) = T(0) = 0.

    but since T is linear:

    0 = T(c1v1 + c2v2 +...+ cnvn) = c1T(v1) + c2T(v2) +...+ cnT(vn)

    which shows that {T(v1),T(v2),...,T(vn)} is linearly dependent.

    it is NOT true that the image of a linearly independent set under a linear transformation is linearly independent.

    for example, if T is the 0-map, T(v) = 0, for all v in V, then even if B = {v1,v2,...,vn} is a basis,

    T(B) = {T(v1),T(v2),...,T(vn)} = {0,0,...,0} = {0} (since repeated elements of a SET don't "count extra"),

    and {0} is ALWAYS a linearly dependent set.

    some equivalent (sufficient) conditions for T(S) to be LI, when S is LI:

    a) T is injective
    b) det(T) ≠ 0
    c) ker(T) = {0}
    d) rank(T) = dim(V) (where V is the domain of T)

    note these conditions are not necessary, it may well be that T(S) is LI when S is, even if none (thus all) of the above hold. for example, let T be multiplication by the matrix:

    [1 0 0]
    [0 1 0]
    [0 0 0].

    then for S = {(1,0,0),(0,1,0)}, T(S) is LI, even though T is singular. however, S' = {(0,1,0),(0,2,2)} is also LI, but T(S) = {(0,1,0),(0,2,0)}, which is NOT linearly independent, since the second vector is a scalar multiple of the first:

    we have 2(0,1,0) + (-1)(0,2,0) = (0,0,0) and neither of {2,-1} is 0.
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