Linear Algebra – Linearity of the Right Inverse of a Surjective Linear Map

Question

Suppose we have a surjective linear map $f:V\to V$ on an infinite-dimensional vector space $V$. We know that every surjective map has at least one right inverse. So I was wondering… I know not all right inverses are linear, for example, on the space of real sequences
$$(a_0,a_1,a_2,\cdots)\mapsto(1,a_0,a_1,a_2,\cdots)$$
is a right inverse of the surjective linear map
$$(a_0,a_1,a_2,\cdots)\mapsto(a_1,a_2,\cdots)$$ but obviously isn't a linear map itself, but on the other hand $$(a_0,a_1,a_2,\cdots)\mapsto(0,a_0,a_1,a_2,\cdots)$$ is a linear right inverse. So I was wondering… does there always exist a linear right inverse of any linear surjective map on an infinite-dimensional vector space over an arbitrary field $K$?

Answer 1

If you write $V=\ker f\oplus W $ (which can always be done), then the restriction $f:W\to V $ is bijective, and you can use its inverse as your linear right-inverse.