# What does it mean for a vector to be “reduced” with respect to a subspace?

Let $V$ be an inner product space with some chosen (not orthogonal in general) basis, $v \in V$ an arbitrary vector, and $U \subset V$ a subspace. Let's further suppose that $U$ is represented by a matrix (which I will also denote $U$) whose columns for a basis in terms of the chosen basis for $V$.

If $v \in U$, then we can write $v = Uw$ for some $w$. If it is not, I have seen the terminology that one can "reduce" $v$ with respect to $U$, that is, write $v = Uw + v'$. What exactly are the conditions on $v'$ that makes this representation "reduced"? Obviously $v'$ should be zero if $v \in U$, but I'm not sure what the requirements are if it is not.

Here are a few possibilities:

1. We're in an inner product space, so take the difference of $v$ with its orthogonal projection onto $U$ (although we might not already have an orthogonal basis for $U$, and because my application is in computing, I'd prefer not to do this).
2. Extend the basis for $U$ to a basis $U'$ for $V$ in some "minimal" way, and compute $v=U'w$, splitting up $w$ appropriately afterward.
3. Since we are in a normed vector space, require some sort of norm minimization on $v'$

EDIT: my main question is: how does one compute this reduction? For the sake of full disclosure, the place I'm seeing this notation is in this paper of Gunnar Carlsson, first paragraph of the second page of section 4. (Algorithm), "case $f_i$".

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I'd guess 1. and 3. (which are equivalent). "Reduction" can mean "make shorter" as in "lattice basis reduction algorithm". – WimC Dec 2 '12 at 17:34