The core principle establishes the criterion under which a set of vectors in a vector space over a field is termed linearly independent, requiring that the only scalar combination yielding the zero vector be the trivial solution where all coefficients vanish. This formal definition relies strictly on axiomatic properties including closure under addition and scalar multiplication, distinguishing between subsets with non-trivial dependencies and those forming bases for subspaces within abstract algebraic structures. The concept functions as a foundational axiom in linear theory, defining the dimensionality of vector spaces independent of any specific coordinate representation or computational basis.
Defines linear independence: no vector in the set is a linear combination of the others, equivalently only the trivial combination gives the zero vector.