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Let $X$ be a finite dimensional vector space and let $X = X_1 \oplus \cdots \oplus X_r$ where $X_i$ is a subspace of $X$. Let $K$ be some subspace of $X$. What can we say about the structure of $K$ in terms of the subspaces $X_i$?

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    Well, $\mathbf R^2 = \mathbf R(1, 0) \oplus \mathbf R(0, 1)$. What would you want your analysis to say about $K = \mathbf R(1, 1)$ here?2012-01-02
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    @Dylan: So the answer is that we can not say anything in general?2012-01-02
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    Direct sums find greater use under the space of transformations from V to V - written Hom(V,V). That's because if V is the direct sum of X and Y then $ Hom(V,V) = Hom(X,X) \oplus Hom(X,Y) \oplus Hom(Y,X) \oplus Hom(Y,Y) $ - an effective 'splitting' of Hom(V,V) into as close as disjoint vector spaces possible. Generally direct sums aren't amazingly useful by themselves.2012-01-02
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    It's not so clear what you mean by "say about." What do you want to know? The image of $K$ under projections? Whether $K$ is isomorphic to some $\oplus_i X_i$? Whether $K = \oplus_i X_i$ for several $i$?2012-01-02
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    @Neal: Yes something like that. To give you an example, i was considering a linear map $B:U \rightarrow X$ and i wanted to have something like $R(B)=K_1 \oplus \cdots \oplus K_r$.2012-01-02
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    @Adam: This is very interesting. Could you please put it as an answer?2012-01-02
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    Fortunately, this is easy to prove despite it's appearance. First you need to recall that two subspaces are direct sums if they add to the original space and are (vector space) disjoint - so {0} is the only common vector of both. You then want to prove Hom(X,X) and Hom(X,Y) are the direct sum of Hom(X,V) - by symmetry this also deals with Hom(Y,X) and Hom(Y,Y), which is the direct sum of Hom(Y,V). In turn they sum to Hom(V,V) then prove their intersection equals {0}. I added my reply here so I can describe the reult with little clutter as an answer.2012-01-02

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Basically you are composing projections with restrictions.

Let f be in Hom(V,V).

My TEXing is weak but this is the idea;

f = (projection of X composed with the restriction of f to X) $ \oplus$ (projection of X composed with the restriction to X) $\oplus$ (projection of X composed with the restriction of Y) $\oplus $(projection of Y composed with the restriction of f to Y).