Subspace as a Direct Sum of Intersections with Basis Partition?

[imurme8]

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

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

 

[jgens]

Notice that V = \bigoplus_{i=1}^k \langle B_i \rangle, so S = S \cap V = \bigoplus_{i=1}^k S \cap \langle B_i \rangle.

 

[imurme8]

Notice that V = \bigoplus_{i=1}^k \langle B_i \rangle, so S = S \cap V = \bigoplus_{i=1}^k S \cap \langle B_i \rangle.

Where have you used that S\cap \langle B_i\rangle \neq \{0\} for all i? I can come up with the following counterexample if we do not assume this hypothesis:

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