## Projections (Pt. II)

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

*Combinations of Projections*

A clear question one may ask about projections is when the sum of two projections is itself a projection. This and more is answered by the next theorem:

**Theorem: ***Let be a finite dimensional -space with . Suppose then that are such that*

*and is the projection on along for . Then,*

**Proof:**

: Suppose first that was a projection. Note then that

and so upon subtraction we find that . But, multiplying this on the left by gives that

and multiplying the same initial equation on the right gives

and equating these gives us that

but we know that

from where it follows that .

Conversely, t if then the above argument shows that is idempotent and thus a projection. Thus, it suffices to prove that it’s a projection on along . It’s clear from our previous theorem that where and . Thus, it suffices to find and . We note firstly that if with where and . We note then that

where we’ve used the fact that and similarly for . We note though that and from where it follows (from the previous theorem) that . The reverse inclusion is clear, and thus it follows that .

Now, it suffices to find . It’s evident that . Conversely, if then we know that

multiplying this equation by on the right (remembering our assumptions) gives that

and multiplying on the right by gives

from where it follows that and thus it follows that . The conclusion follows.

: We merely note that by our previous theorem is a projection if and only if

is a projection. But, by of this theorem we know that this is true if and only if

but clearly upon expansion this is true if and only if . Moreover, if

we have from the previous problem that it is the projection on along .

**References:**

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

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