# Abstract Nonsense

## Projections (Pt. I)

Point of post: In this post we discuss the concepts of projections onto subspaces of a vector space.

Motivation

It makes sense that if the internal direct sum of subspaces of a vector space should have any significance in the overall theory, then so should the projections onto theses subspaces. The idea of a projection is that if $\mathscr{V}=\mathscr{U}\oplus\mathscr{W}$ with $\mathscr{U},\mathscr{W}\leqslant \mathscr{V}$ then for every $v\in\mathscr{V}$ the projection along $\mathscr{U}$ should be a function which takes in $v$ and gives back the $\mathscr{U}$ part.

Projections

Let $\mathscr{V}$ be a $F$-space and $\mathscr{U},\mathscr{W}\leqslant \mathscr{V}$ be such that $\mathscr{V}=\mathscr{U}\oplus\mathscr{W}$. We define the projection on$\mathscr{U}$ along $\mathscr{W}$, denoted $\pi_{\mathscr{U}}$ (where the $\mathscr{W}$ is clear from context), to be the function

$\text{ }$

$\pi_{\mathscr{U}}:\mathscr{V}\to\mathscr{U}:u+w\mapsto u$

$\text{ }$

where $u+w$ is the unique representation of any vector in $\mathscr{V}$. It’s evident that $\pi_{\mathscr{U}}\in\text{End}\left(\mathscr{V}\right)$ but for $\mathscr{U}\ne \mathscr{V}$ we have that $\pi_{\mathscr{U}}\notin\text{GL}\left(\mathscr{V}\right)$. It’s clear though that a projection on $\mathscr{U}$ is, in terms of the function, independent (value wise) of which complement of $\mathscr{U}$ $\pi_{\mathscr{U}}$ is along.

Our first theorem says that projections can really be described entirely by a single property. Namely, if $T\in\text{End}\left(\mathscr{V}\right)$ we call $T$ idempotent if $T^2=T$.

Theorem: Let $\mathscr{V}$ be a finite dimensional $F$-space and $T\in\text{End}\left(\mathscr{V}\right)$. Then, $T$ is a projection along some subspace if and only if $T$ is idempotent.

Proof: Clearly if $T=\pi_{\mathscr{U}}$ with $\mathscr{V}=\mathscr{U}\oplus\mathscr{W}$ then evidently for any $u+w\in\mathscr{V}$ we have that

$\text{ }$

$\pi_{\mathscr{U}}\left(\pi_{\mathscr{U}}\left(u+w\right)\right)=\pi_{\mathscr{U}}\left(u\right)=u=\pi_{\mathscr{U}}\left(u+w\right)$

$\text{ }$

and thus $\pi_{\mathscr{U}}^2=\pi_{\mathscr{U}}$.

Conversely, suppose that $T^2=T$. Let $\mathscr{U}=\left\{v\in V:T(v)=v\right\}$ and $\mathscr{W}=\ker T$. We claim that $\mathscr{V}=\mathscr{U}\oplus\mathscr{W}$. Indeed, if $v\in\mathscr{U}\cap\mathscr{W}$ then $\bold{0}=T(v)=v$. Also, let $v\in\mathscr{V}$ be arbitrary and note that $v=T(v)+(v-T(v))$ and

$\text{ }$

$\displaystyle T(T(v))=T(v)$

$\text{ }$

and

$\text{ }$

$T\left(v-T(v)\right)=T(v)-T(T(v))=T(v)-T(v)=\bold{0}$

$\text{ }$

and thus it follows that $v\in\mathscr{U}$ and $v-T(v)\in\mathscr{W}$ from where it follows that $\mathscr{V}=\mathscr{U}+\mathscr{W}$. Now, we claim that $T$ is the projection on $\mathscr{U}$ along $\mathscr{W}$. Indeed, for every $u+w\in\mathscr{V}$ with $u\in \mathscr{U}$ and $w\in\mathscr{W}$ we have that

$\text{ }$

$T\left(u+w\right)=T(u)+T(w)=u+\bold{0}=u$

$\text{ }$

from where the conclusion follows. $\blacksquare$

Next, we prove that the apparent asymmetry in the definition of a projection on $\mathscr{U}$ along $\mathscr{W}$ is unjustified, in the sense that every such projection induces a projection with the roles of $\mathscr{U}$ and $\mathscr{W}$. More formally:

Theorem: Let $\mathscr{V}$ be a finite dimensional $F$-space with $\mathscr{V}=\mathscr{U}\oplus\mathscr{W}$. Then, if $\pi_{\mathscr{U}}$ is the projection on $\mathscr{U}$ along $\mathscr{W}$ then $\mathbf{1}-\pi_{\mathscr{U}}$ is the projection on $\mathscr{W}$ along $\mathscr{U}$.

Proof: This is evident from the simple calculation that for any $u+w\in\mathscr{V}$ we have that

$\text{ }$

$\left(\mathbf{1}-\pi_{\mathscr{U}}\right)(u+w)=\mathbf{1}(u+w)-\pi_{\mathscr{U}}\left(u+w\right)=u+w-u=w$

$\text{ }$

from where the conclusion follows. $\blacksquare$

From this we get the notion that if $\pi_{\mathscr{U}}$ is the projection of $\mathscr{U}$ along $\mathscr{W}$ and $\pi_{\mathscr{W}}$ the projection on $\mathscr{W}$ along $\mathscr{U}$ then $\pi_{\mathscr{W}}=\mathbf{1}-\pi_{\mathscr{U}}$, or said differently $\mathbf{1}=\pi_{\mathscr{U}}+\pi_{\mathscr{W}}$.

References:

1. Halmos, Paul R.  Finite-dimensional Vector Spaces,. New York: Springer-Verlag, 1974. Print

January 13, 2011 -

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