Generating Sets and Bases 2.9 2001/03/15 2002/10/24 Thanos Antoulas aca@rice.edu John Paul Slavinsky jps@alumni.rice.edu Roy Ha rha@rice.edu Elizabeth Chan lizychan@rice.edu Thanos Antoulas aca@rice.edu John Paul Slavinsky jps@alumni.rice.edu vector space generating set basis bases This module defines generating sets and bases in linear algebra. Given x 1 … x k , we can define a linear space (vector space) X as X span x 1 … x k i 1 k α i x i α i In this case, x 1 … x k form what is known as a generating set for the space X. That is to say that any vector in X can be generated by a linear combination of the vectors x 1 … x k . If x 1 … x k happen to be linearly independent, then they also form a basis for the space X. When x 1 … x k define a basis for X, k is the dimension of X. A basis is a special subset of a generating set. Every generating set includes a set of basis vectors. The following three vectors form a generating set for the linear space 2 . x 1 1 1 , x 2 1 0 , x 3 2 1 It is obvious that these three vectors can be combined to form any other two dimensional vector; in fact, we don't need this many vectors to completely define the space. As these vectors are not linearly independent, we can eliminate one of them. Seeing that x 3 is equal to x 1 x 2 , we can get rid of it and say that our basis for 2 is formed by x 1 and x 2 .