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0.12 Generating sets and bases

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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 i x i 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 generatingset 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 thismany 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 .

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Source:  OpenStax, State space systems. OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10143/1.3
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