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Generating Sets and Bases

Module by: Thanos Antoulas, JP Slavinsky

Summary: This module defines generating sets and bases in linear algebra.

Given x1…xk∈ℝ x 1 … x k , we can define a linear space (vector space) XX as

X=spanx1…xk={∑i=1k α i xi| α i ∈ℝ} X span x 1 … x k i 1 k α i x i α i (1)
In this case, x1…xk x 1 … x k form what is known as a generating set for the space XX. That is to say that any vector in XX can be generated by a linear combination of the vectors x1…xk x 1 … x k .

If x1…xk x 1 … x k happen to be linearly independent, then they also form a basis for the space XX. When x1…xk x 1 … x k define a basis for XX, kk is the dimension of XX. A basis is a special subset of a generating set. Every generating set includes a set of basis vectors.

Example 1

The following three vectors form a generating set for the linear space ℝ2 2 .

x1=11 x 1 1 1 , x2=10 x 2 1 0 , x3=21 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 x3 x 3 is equal to x1+x2 x 1 x 2 , we can get rid of it and say that our basis for ℝ2 2 is formed by x1 x 1 and x2 x 2 .

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