# Example of Projection

1. Nov 2, 2009

### SpringPhysics

1. The problem statement, all variables and given/known data
Give an example of a subspace W of a vector space V such that there are two projections on W along two distinct subspaces.

2. Relevant equations

3. The attempt at a solution
I tried looking into Euclidean geometry spaces (R3 and R2) but no matter what subspace W I choose, there is only one subspace along which W projects. For example, if my vector space is (x,y,z) and my subspace W is (x,y,0), then by the properties of subspaces in projection, the other subspace must be (0,0,z). How is it possible to get two distinct subspaces along which W projects, and still have a direct sum of the same vector space?

2. Nov 3, 2009

### HallsofIvy

Staff Emeritus
What? One of us is completely misunderstanding the problem. You said "Give an example of a subspace W of a vector space V such that there are two projections on W along two distinct subspaces." Let V= R3, W= {(x,y,0}). Then the projections (x,y,z)->(x, 0, 0) and (x,y,z)-> (0, y, 0) are projections on W along different subspaces. The orthogonal complement of W, (0, 0, z) has nothing to do with the problem.

3. Nov 3, 2009

### SpringPhysics

I thought that by definition, projection only works if V (vector space) = W (one subspace) (+) W' (another subspace) [i.e. V is the direct sum of two subspaces] - so when the question asks for two projections on W along two distinct subspaces, wouldn't the "distinct subspaces" each have to add to W to yield V?