Suppose U and V are finite-dimensional subspaces of some finite-dimensional space W (over the field Q in my actual case, but probably irrelevant)(adsbygoogle = window.adsbygoogle || []).push({});

I've got a subspace in mind that I can't quite define well. I'll start with a concrete example:

Consider R^3 with orthonormal basis {x, y, z}. Let U = Span{x, y} and V = Span{y, z}. The symmetric difference of {x, y} and {y, z} is {x, z}, and the strange thing I want is Span{x, z}. I resorted to using an "obvious" basis set here, but I want to know if the thing I'm looking for is well-defined. What if I haven't picked a basis; what if it's not a normed space; what if it's not even an inner product space; etc.

A bit more formally: Let [tex]U \cap V = I[/tex] have dimension k > 0. Then I can pick an orthonormal basis for I, Bi = {b1, ..., bk}, and then choose an orthonormal basis for [tex]U + V = Span(U \cup V)[/tex] (call it simply B) that has Bi as a subset. Now Span(B \ Bi) is uniquely defined (I think).

Is that well-defined, or is my intuition wrong? If it is, is there a cleaner way of specifying it?

Even if this is true, my problems have only just begun: the W I have in mind is not an inner product space; it's R over Q, with U and V being subspaces of some n-dimensional subspace S.

S is isomorphic to Q^n, so I can define an inner product and norm, but this seems like a painful way to go about things.

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# A strange kind of subspace

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