# Proof regarding orthogonal projections onto spans

## Homework Statement

Let U be the span of k vectors, {u1, ... ,uk} and Pu be the orthogonal projection onto U. Let V be the span of l vectors, {v1, ... vl} and Pv be the orthogonal projection onto V. Let X be the span of {u1, ..., uk, v1, ... vl} and Px be the orthogonal projection onto X.

Show Px*y = Pu*y + Pv*y if and only if the space U is orthogonal to the space V (for all y in R^n).

I'm having trouble on both sides of this if and only if proof. Any help? thanks.

See above

## The Attempt at a Solution

I'm a bit lost - this seems intuitive but i'm having trouble processing it...