## Projections (Pt. III)

**Point of post: **This is a continuation of this post.

: It’s clear that if then is a projection since

We use the same methodology to find the and with which is on and along respectively. We claim that . Indeed, if then

and so . The reverse inclusion is clear from where it follows that .

Now we next claim that . Indeed, if then we have that

and thus and . We note though that

and since it follows that . The revere inclusion is evident. The conclusion follows.

*Projections and Invariances*

Lastly we discuss how projections tie into the invariance of subspaces. Namely, we have the following theorem:

**Theorem: ***If is a finite dimensional vector space and is invariant under then for any projection on . Conversely, if for some projection on along then is invariant under .*

**Proof: **Suppose first that is invariant under and is a projection on on any complement of . We see then that (remembering that )

and since was arbitrary it follows that .

Conversely, if then for any we have that

but this is true if and only if . It follows from the arbitrariness of that is invariant under .

Our next theorem gives us a similar formulation for reducibility of a linear transformation in terms of projections. Namely:

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

**Proof: **Suppose first that and reduce then we have that that

Conversely, suppose that it’s evident from the last theorem that is invariant under since

But, is also invariant under . Indeed, let . Then,

from where it follows that . Since was arbitrary the conclusion follows.

The proof of is continued here.

**References:**

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

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