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]?(adsbygoogle = window.adsbygoogle || []).push({});

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

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

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