Linear independence and span constitute fundamental pillars within the theory of vector spaces in linear algebra, defining the structural properties necessary for analyzing dimensionality and subspace relationships. The core principle establishes that a set of vectors is independent if no non-trivial linear combination yields the zero vector, while spanning defines the capacity to generate every vector in the space through such combinations. These abstract concepts provide the rigorous definitions required to determine basis sets and calculate the intrinsic dimensions of finite-dimensional spaces over arbitrary fields.
Works an example determining whether a set of vectors spans R^3 and whether the set is linearly independent, connecting span and independence.