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# Generating set of a vector space.

Here too, we consider a vector space V and a set I (this will be a set of indices, generally of the form , but not necessarily).

Definition 5.3.1   A family of vectors in V is a generating set for V if every vector in V is a linear combination of the 's, i.e.

Example 5.3.2   We prove that the vectors and form a generating set for .

Let . Then:

Thus, any vector in is a linear combination of and .

Proposition 5.3.3   Let F be a generating set for the vector space V. If G is a family of vectors such that , then G is a generating set for V.

Hint: Add zeros as coefficients for the extra vectors.

Noah Dana-Picard
2001-02-26