Is a subspace the direct sum of all its intersections with a partition of the basis?

  • Thread starter imurme8
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I've been working on this Linear Algebra problem for a while: Let [itex]F[/itex] be a field, [itex]V[/itex] a vector space over [itex]F[/itex] with basis [itex]\mathcal{B}=\{b_i\mid i\in I\}[/itex]. Let [itex]S[/itex] be a subspace of [itex]V[/itex], and let [itex]\{B_1, \dotsc, B_k\}[/itex] be a partition of [itex]\mathcal{B}[/itex]. Suppose that [itex]S\cap \langle B_i\rangle\neq \{0\}[/itex] for all [itex]i[/itex]. Is it true that [itex]S=\bigoplus\limits_{i=1}^{k}(S\cap \langle B_i \rangle)[/itex]?

Haven't been able to get this one, thanks for your help.
 

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jgens
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Notice that [itex]V = \bigoplus_{i=1}^k \langle B_i \rangle[/itex], so [itex]S = S \cap V = \bigoplus_{i=1}^k S \cap \langle B_i \rangle[/itex].
 
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Notice that [itex]V = \bigoplus_{i=1}^k \langle B_i \rangle[/itex], so [itex]S = S \cap V = \bigoplus_{i=1}^k S \cap \langle B_i \rangle[/itex].
Where have you used that [itex]S\cap \langle B_i\rangle \neq \{0\}[/itex] for all [itex]i[/itex]? I can come up with the following counterexample if we do not assume this hypothesis:

In [itex]\mathbb{R}^2[/itex], the subspace [itex]y=x[/itex] is certainly not the direct sum of its intersections with [itex]\langle e_1 \rangle[/itex] and [itex]\langle e_2 \rangle[/itex] (both zero).
 

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