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

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To prove the existence of a linear mapping L: V → V with kernel S, one effective approach is to utilize the orthogonal projection onto the complement of S within the finite-dimensional vector space V. This method leverages the properties of linear transformations and subspaces. The orthogonal projection ensures that all vectors in S are mapped to zero, thus establishing S as the kernel of L. Understanding this relationship between subspaces and linear mappings is crucial for constructing the proof. This approach provides a solid starting point for further exploration and formal proof development.
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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!
 
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Take the orthogonal projection to the complement subspace of S.
 

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