Proving Existence of Linear Mapping with Kernel in Subspace S | Helpful Guide

[Bazzinga]

Hey guys, I was wondering if you could help me out with a question I've got, I really don't know where to go or really where to start! Here's the question:

Let S be a subspace of a finite dimensional vector space V. Show that there exists a Linear Mapping L: V → V such that the kernel of L is S.

I started off messing around with some examples and the theorem makes sense to me, I just can't figure out how to prove it! If someone could start me off that would be awesome.

Thanks!

 

[csopi]

Take the orthogonal projection to the complement subspace of S.