Example: Projection Subspaces: Solving a Challenging Homework Statement

[SpringPhysics]

Homework Statement

Give an example of a subspace W of a vector space V such that there are two projections on W along two distinct subspaces.

Homework Equations

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?

 

[HallsofIvy]

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.

 

[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?