Proof regarding orthogonal projections onto spans

[samuelr0750]

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.

Homework Equations

See above

The Attempt at a Solution

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

 

[brmath]

Just to get some insight, start with a simple case. Take R3 as your underlying space and let U and V each be 1 dimensional. So each is a line. Pu is on the U line and Pv is on the V line. What is the subspace X spanned by the basis vectors of U and V? ? If y is in X can you see why the theorem would be true in this case? Consider 3 possibilities: y is in U, y is in V or y is in neither.

To generalize, consider the composition of y in X. The u's and v's are the basis of X, so you can express y in terms of those basis vectors. If you do that, and think about the situation in the above paragraph, perhaps you can get started.