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Definition of a span and generators?
What does it mean by span in terms of linear algebra? I don t see how it differs from a basis
Also, I need to know what a "system of generators" are in the context of linear algebra also. I can t find a clear definition anywhere.
1 Answer
- husoskiLv 73 years ago
The span of a set of vectors is the set of all linear combination of vectors in the set.
A basis for a vector space is a *linearly independent* set of vectors that spans the whole space.
A generating set for a vector space is any set of vectors that span the space. I guess "system of generators" is another name for that set.
So, a basis is a generating set where the vectors are linearly independent.
For example, you could take the set of vectors from R²: G = {(1, 0), (1, 1) and (1, 1)}. The span of the vectors in G are all of R², so this is a generating set (system of generators) for R². But (1,1) = (1,0) + (0,1), so the set is not linearly independent and not a basis.
