# Abstract Nonsense

## 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.

Motivation

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 $f:X\to Y$ where $X$ is Hausdorff and $Y$ is compact is continuous if and only if the graph is closed in $X\times Y$ 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 $\mathscr{V}$ and $\mathscr{W}$ are $F$-spaces and $T:\mathscr{V}\to\mathscr{W}$ we may define the graph, denotes $\Gamma_T$, to be the set $\left\{\left(v,T(v)\right):v\in\mathscr{V}\right\}$. Then, $T\in\text{Hom}\left(\mathscr{V},\mathscr{W}\right)$ if and only if $\Gamma_T$ is a subspace of $\mathscr{V}\boxplus\mathscr{W}$.

So, that being said, all I have left to say is the proof of this interesting theorem:

Theorem: Let $\mathscr{V}$ and $\mathscr{W}$ be $F$-spaces and $T:\mathscr{V}\to\mathscr{W}$. Then, $T\in\text{Hom}\left(\mathscr{V},\mathscr{W}\right)$ if and only if $\Gamma_T$ is a subspace of $\mathscr{V}\boxplus\mathscr{W}$.

Proof: If $T$ is a linear transformation then we evidently see that

$\alpha\left(v,T(v)\right)+\beta\left(v',T(v')\right)=\left(\alpha v+\beta v',\alpha T(v)+\beta T(v')\right)=\left(\alpha v+\beta v',T\left(\alpha v+\beta v'\right)\right)$

so that $\Gamma_T$ is indeed a subspace.

Conversely, if $\Gamma_T$ is a subspace of $\mathscr{V}\boxplus\mathscr{W}$, $v,v'\in\mathscr{V}$ and $\alpha,\beta\in F$ then we note that since $\left(v,T(v)\right),\left(v',T\left(v'\right)\right)\in\Gamma_T$ and $\alpha,beta\in F$ that $\alpha(v,T(v))+\beta (v',T(v'))\in\Gamma_T$. But since the first coordinate of $\alpha(v,T(v))+\beta (v',T(v'))$ is $\alpha v+\beta v'$ and $\alpha(v,T(v))+\beta(v',T(v'))\in\Gamma_T$ it follows that the second coordinate must equal $T\left(\alpha v+\beta v'\right)$. But, by definition we have that

$\alpha(v,T(v))+\beta(v',T(v'))=\left(\alpha v+\beta v',\alpha T(v)+\beta T(v')\right)$

and thus it follows that

$T\left(\alpha v+\beta v'\right)=\alpha T(v)+\beta T(v')$

from where the conclusion follows. $\blacksquare$

References:

1. Golan, Jonathan S. The Linear Algebra a Beginning Graduate Student Ought to Know. Dordrecht: Springer, 2007. Print.