I just come up with a question that I usually ignore. Let $T:V\longrightarrow W$ be a linear transformation. Then $T$ may or may not have a left inverse (right, resp.). But if $T$ has a left inverse (right, resp.) $f$, then must $f$ be a linear transformation?
2026-04-13 21:44:12.1776116652
Must every left inverse function of a linear transformation be a linear transformation again?
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No. Take $T\colon\mathbb{R}^2\longrightarrow\mathbb R$ defined by $T(x,y)=x+y$. Define $f\colon\mathbb{R}\longrightarrow\mathbb{R}^2$ by $f(x)=(x+1,-1)$. Then $f$ is not linear, but $(\forall x\in\mathbb{R}):T\bigl(f(x)\bigr)=x$, which means that $f$ is a right inverse of $T$.