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Math Help - Algebra, Problems For Fun (28)

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    Algebra, Problems For Fun (28)

    Definition: Let V be a vector space. A linear transformation T: V \longrightarrow V is left (right) invertible if ST=\text{id}_V ( TS=\text{id}_V), for some linear transformation S: V \longrightarrow V.

    Problem: Give an example of a vector space V and a linear transformation T: V \longrightarrow V such that T is left but not right invertible.
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    MHF Contributor Bruno J.'s Avatar
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    There are no such transformations between finite dimensional vector spaces : because then if T is left invertible, then T is injective, and hence surjective, and hence has both a left and a right inverse.

    Consider the vector space V consisting of sequences of elements (x_i)_{i\in \mathbb{N}} in some field. Let T be the "push" operator mapping (x_1,x_2,...) \mapsto (0,x_1,x_2,...) . Then T is clearly an injection, hence left invertible, but clearly is not a surjection, hence is not right invertible.
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