For a given field F let Lambda_{F} be the category of finite-dimensional vector spaces over F. Show that the functor from Lambda_{F}^{op} to Lambda_{F} that assigns to each vector space V its dual V^{*} and to each linear map phi its dual (sometimes called its transpose) is an exact functor.
If F is a field and V, W are finite-dimensional vector spaces over F, prove that there is a natural isomorphism
Recall that a short exact sequence of R-modules is said to be split if it arises from a direct sum decomposition of its middle term. Let
be a short exact sequence of R-modules. Note that since the map M -----> M'' is surjective it must have a set-theoretic right inverse and since the map M' -----> M is injective it must have a set-theoretic left inverse.
Prove that the exact sequence is split if and only if the linear map M -----> M'' admits an R-linear right inverse.
Prove that the exact sequence is split if and only if the linear map M' -----> M admits an R-linear left inverse.
Let R be a commutative ring and I, J ideals in R. Prove that
where I + J denotes the smallest ideal in R containing the set I \cup J.
Let Gamma be a cateory. If A is an object in Gamma and X is an object in Gamma, the set Hom_{Gamma}(A, X) is called the set of ``X-valued points of A'', and the functor F_{A} given by F_{A}(X) = Hom_{Gamma}(A, X) is called the ``functor of points of A''. Explain what it means mean for F_{A} and F_{B} to be isomorphic, and then show that F_{A} and F_{B} are isomorphic if and only if A and B are isomorphic in Gamma.
Note: There are contexts where it makes sense to regard Hom(X, A), instead of Hom(A, X), as the set of X-valued points of A. This is equivalent to regarding A as an element of the dual category and taking its X-valued points in that category.