Linear Independence and Basis

The Minimal Building Blocks

A linear combination of vectors $v_1, \dots, v_k$ is any sum $c_1 v_1 + \dots + c_k v_k$ with scalar coefficients. The set of all linear combinations of some vectors is their span — the whole region they can reach. In the plane, the span of two non-parallel vectors is the entire plane; the span of a single vector is just a line.

Vectors are linearly independent if none of them is a linear combination of the others — equivalently, the only way to write $c_1 v_1 + \dots + c_k v_k = \mathbf{0}$ is with all coefficients zero. If some nontrivial combination gives zero, the vectors are dependent — one is redundant, expressible from the others. Independence means "no wasted vectors."

A basis for a vector space is a set of vectors that is both:

• Linearly independent (no redundancy), and
• Spanning (their combinations reach every vector in the space).

A basis is a minimal spanning set — just enough vectors, with none wasted, to build everything. The number of vectors in a basis is the dimension of the space (the plane has dimension 2, ordinary space dimension 3). Every vector has a unique expression in a given basis — the essence of coordinates. This ties everything together: coordinates, determinants (nonzero means the columns are independent, i.e. a basis), and solving systems.

Common pitfall: confusing spanning with independence, or thinking any spanning set is a basis. Spanning means the vectors reach everything (possibly with redundancy); independence means no redundancy (but possibly not reaching everything). A basis needs both — a spanning set with extra, dependent vectors is not a basis, and neither is an independent set too small to span.