Week2 - Supplement to Week 2 0.1 Terminology Let A and B be...

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Supplement to Week 2 January 26, 2007 0.1 Terminology Let A and B be sets. Recall that A B means the set of all functions f : A B from A to B . If f is such a function, the set A is called the domain of f and the set B is called the codomain of f . The set of all b B such that there exists an a A such that f ( a ) = b is called the image of f , or (sometimes) the range of f . In general, the image of f is a subset of B , and f is said to be surjective (“onto”) if the image of f is all of B . Thus, f is surjective if for each b B , there is a t least one a A such that f ( a ) = b . On the other hand, f is said to be injective (“one-to-one”) if for each b B , there is at most one a A such that f ( a ) = b . Finally, f is said to be bijective if it is both injective and surjective. In this case it has a well-defined inverse function f - 1 : B A . 0.2 Linear combinations It is important to have a way to discuss the process of forming linear com- binations in a precise way. It is most convenient to do this with sequences

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