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    Linear transformations

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    Let L: V -- W be a linear transformation, and let T be a subspace of W. The inverse image of T denoted L^-1(T), is defined by L^-1(T) = {v e V | L(v) e T}. Show that L^-1(T) is a subspace of V.

    A linear transformation L: V -- W is said to be one-to-one if L(v1) = L(v2) implies that v1=v2. Show that L is one-to-one if and only if ker(L) = {0v}.

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    Solution Preview

    First part:
    T is a subspace, so 0 e T, moreover L is linear i.e. L(0)=0, so for 0 e T ; 0v= L^-1(0T) e L^-1(T)

    Now we suppose that y1 and y2 e L^-1(T) therefore we can define this as {y1 e V | L(y1) e T} and {y2 e V | L(y2) e T} respectively. Now we consider ...

    Solution Summary

    The solution shows how to prove that an inverse transformation is a subspace, and how to prove a transformation is one-to-one if and only if a certain condition is met.