Linear Independence, Span, and Bases
2.1 Span and Linear Independence
Let V be a vector space over a field F.
A linear combination of the vectors v1, v2, ..., vk ∈ V is a sum of scalar multiples of these vectors; that is, c1v1 + c2v2 + ・・・ + ckvk, for some scalar coefficients c1, c2, ..., ck ∈ F. If S is a set of vectors in V, a linear combination of vectors in S is a vector of the form c1v1 + c2v2 + ・・・ + ckvk with k ∈ ℕ, vi ∈ S, ci ∈ F. Note that S may be finite or infinite, but a linear combination is, by definition, a finite sum. The zero vector is defined to be a linear combination of the empty set.
When all the scalar coefficients in a linear combination are 0, it is a trivial linear combination ...