## 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 with then for every the *projection along *should be a function which takes in and gives back the * part*.

*Projections*

Let be a -space and be such that . We define the *projection on along *, denoted (where the is clear from context), to be the function

where is the unique representation of any vector in . It’s evident that but for we have that . It’s clear though that a projection on is, in terms of the function, independent (value wise) of which complement of is along.

Our first theorem says that projections can really be described entirely by a single property. Namely, if we call *idempotent *if .

**Theorem: ***Let be a finite dimensional -space and . Then, is a projection along some subspace if and only if is idempotent.*

**Proof: **Clearly if with then evidently for any we have that

and thus .

Conversely, suppose that . Let and . We claim that . Indeed, if then . Also, let be arbitrary and note that and

and

and thus it follows that and from where it follows that . Now, we claim that is the projection on along . Indeed, for every with and we have that

from where the conclusion follows.

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

**Theorem: ***Let be a finite dimensional -space with . Then, if is the projection on along then is the projection on along .*

**Proof: **This is evident from the simple calculation that for any we have that

from where the conclusion follows.

From this we get the notion that if is the projection of along and the projection on along then , or said differently .

**References:**

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

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

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

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