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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? – Dylan Moreland Jan 2 '12 at 21:22
@Dylan: So the answer is that we can not say anything in general? – Manos Jan 2 '12 at 21:30
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. – Adam Jan 2 '12 at 21:38
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 $i$? – Neal Jan 2 '12 at 21:39
@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 $R(B)=K_1 \oplus \cdots \oplus K_r$. – Manos Jan 2 '12 at 22:15
up vote 1 down vote accepted

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).

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