I am reading the not yet published book of Eisenbud and Harris about intersection theory (http://isites.harvard.edu/fs/docs/icb.topic720403.files/book.pdf) and I don't quite understand the following: In Proposition 4.31, page 113, they state that $deg(\sigma_a \sigma_b) = 1 $ if $ a_i + b_{k-i+1} = n-k$ for all i and $\sigma_a \sigma_b = 0$ otherwise, where $\sigma_a, \sigma_b$ are the classes of two Schubert cycles of complementary dimension.
1) The part that I don't understand is that they now conclude (in Corollary 4.32) that the Schubert Cycles form a Basis of the Chow Group A(G) of the Grassmannian G.
The "generating-part" is already clear by a different theorem, but I don't see how the "free-part" follows from the proposition above.
2) They also state that the "Schubert Classes form the dual bases of the intersection forms $A^m(G)\times A^{dimG-m} \rightarrow \mathrm Z$". I guess they mean that for a fixed cyle $\sigma_a $ we get an element of the dual of the (codimension m - part of the) Chow Group (viewed as a Z-module) by defining $ \phi_a (\sigma_b) := deg(\sigma_a \sigma_b)$, and IF the $\sigma_a$'s would form a basis, than we had a dual basis that way by the proposition. But still, this does not proof that the Schuber Cycles really form a basis.
Do I get s.th. totally wrong? Am I blind? Thanks in advance.