Abstract Nonsense

Crushing one theorem at a time

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

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January 13, 2011 - Posted by | Algebra, Halmos, Linear Algebra | ,

4 Comments »

  1. […] Point of post: This is a continuation of this post. […]

    Pingback by Projections (Pt. II) « Abstract Nonsense | January 13, 2011 | Reply

  2. […] this fact to study projections in the group algebra which are functions generalizing the notion of projections on an endomorphism algebra. Namely, projections are elements of the group algebra which are idempotent under convolution. […]

    Pingback by Representation Theory: Projections Into the Group Algebra (Pt. I) « Abstract Nonsense | April 9, 2011 | Reply

  3. […] prior observation we know that and it’s coordinate, call it , satisfies and thus by the characterization of projection endomorphism in terms if idempotence we may conclude that is a projection. Suppose that projects onto and . […]

    Pingback by Representation Theory: Projections Into the Group Algebra (Pt. II) « Abstract Nonsense | April 9, 2011 | Reply

  4. […] a ring is one such that (compare to linear projections and projections into the group algebra). The above more generically shows (if one looks hard […]

    Pingback by Ring Homomorphisms (Pt. II) « Abstract Nonsense | June 18, 2011 | Reply


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