Linear Independence and Basis

Spanning and Independence

A linear combination is any sum $c_1 v_1 + \dots + c_k v_k$. Their span is the set of all such combinations—the entire region they can reach.

Vectors are linearly independent if no vector is a combination of the others. Equivalently, the only solution to $c_1 v_1 + \dots + c_k v_k = \mathbf{0}$ is all $c_i = 0$. Dependent means redundancy.

Basis and Dimension

A basis for a vector space is a set of vectors that is linearly independent AND spanning. It is a minimal spanning set with no wasted vectors.

The number of vectors in a basis is the dimension of the space. Every vector has a unique expression in a given basis, which forms the foundation of coordinates.

Common pitfall: Confusing spanning with independence. A basis requires both. A spanning set with extra vectors is not a basis, nor is an independent set too small to span.