Definitions for

**"Vector space"**A set closed under addition and scalar multiplication. One example is R^n, where addition is the usual coordinate-wise addition, and scalar multiplication is t(x1,...,xn) = (tx1,...,txn). Another vector space is the set of all matrices. If A and B are two matrices (of the same size), so is A+B. Also, tA is a matrix for any scalar, t in R. Another vector space is the set of all functions with domain X and range in R^n. If f and g are two such functions, so are f+g and tf for all t in R. Note that a vector space must have a zero since we can set t=0.

a collection of objects that can be added and multiplied by scalars

a flat space that extends out to infinity and has a chosen point in it called the origin or zero