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Thread: Algebra, Problems For Fun (28)

  1. #1
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    Algebra, Problems For Fun (28)

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

    Problem: Give an example of a vector space $\displaystyle V$ and a linear transformation $\displaystyle T: V \longrightarrow V$ such that $\displaystyle T$ 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 $\displaystyle T$ is left invertible, then $\displaystyle T$ is injective, and hence surjective, and hence has both a left and a right inverse.

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