Group items tagged

Lemma Let U and V be vector spaces, and let b:UxV-->X be a bilinear map from UxV to a vector space X. Suppose that for every bilinear map f defined on UxV there is a unique linear map c defined on X such that f=cb. Then there is an isomorphism i:X-->U@V such that u@v=ib(u,v) for every (u,v) in U@V. We can avoid mentioning u@v if we use the map g:UxV-->U@V. Then the lemma says that g=ib. Briefly, the point of the lemma is that any bilinear map b:UxV-->X satisfying the universal property is isomorphic to the map g:UxV-->U@V in an obvious sense.

This coincides with the usual definition of the rank of a matrix. (Exercise: prove this.)

Colleges are responsible for admitting students to Cambridge. A Cambridge College is more than just a hall of residence: it’s a place where you’ll do a lot of living, studying and socialising. Your College community will include undergraduate and research students, teachers and lecturers. But you’ll also make friends with students from other Colleges in Cambridge!

but you will have a room of your own (unless you choose one of the specially-designed shared options).

The personal teaching and support available within the Cambridge Colleges is the most distinctive feature of the collegiate system.

approximate number system (ANS)