# Thread: Algebra, Problems For Fun (28)

1. ## 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.

2. 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.