# Math Help - Algebra, Problems For Fun (28)

1. ## 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.

2. 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.