Interesting Theorem Regarding Linear Transformations
Point of post: In this post we discuss an interesting result which tells us precisely when a mapping from one vector space to another is a linear transformation, namely if and only if the graph (to be defined below) is a subspace of the direct sum of the two vector spaces.
As someone who has done the majority of their mathematical work in topology I can say I am well acquainted with the innocuous concept of the graph of a mapping playing an important role in the theory of structure preserving maps. There is the Closed Graph Theorem in functional analysis, the fact that a function from where is Hausdorff and is compact is continuous if and only if the graph is closed in with the product topology ,etc. That said, I had no idea, until now, that there is a simple but satisfying analogue for linear transformations. Namely, if and are -spaces and we may define the graph, denotes , to be the set . Then, if and only if is a subspace of .
So, that being said, all I have left to say is the proof of this interesting theorem:
Theorem: Let and be -spaces and . Then, if and only if is a subspace of .
Proof: If is a linear transformation then we evidently see that
so that is indeed a subspace.
Conversely, if is a subspace of , and then we note that since and that . But since the first coordinate of is and it follows that the second coordinate must equal . But, by definition we have that
and thus it follows that
from where the conclusion follows.
1. Golan, Jonathan S. The Linear Algebra a Beginning Graduate Student Ought to Know. Dordrecht: Springer, 2007. Print.
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