A property of surjective linear transformations

Mar 21, 2025 3:41 PM

Let $ϕ : V \to W$ be a surjective linear transformation of finite-dimensional linear spaces. Show that there is a $U \subseteq V$ such that $V = (\ker (ϕ)) \oplus U$ and $ϕ ∣_{U} : U \to W$ is an isomorphism. (Note that $V$ is not assumed to be an inner-product space; also note that $\ker (ϕ)$ is sometimes referred to as the null space of $ϕ$ ; finally, $ϕ ∣_{U}$ denotes the restriction of $ϕ$ to $U$ .)

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Let $\phi: V\to W$ be a surjective linear transformation of finite-dimensional linear spaces. Show that there is a $U\subseteq V$ such that $V=(\ker(\phi))\oplus U$ and $\phi\mid_U:U\to W$ is an isomorphism. (Note that $V$ is not assumed to be an inner-product space; also note that $\ker(\phi)$ is sometimes referred to as the {\bfseries null space} of $\phi$; finally, $\phi\mid_U$ denotes the restriction of $\phi$ to $U$.)