# Definition A *basis* is an *ordered* [[Linear Independence|linearly indpendent]] subset of a vector space $\mathcal{B} \subset V$, whose [[Span]] is all of $V$. In simple terms, the basis has enough terms to "make" all of $V$. Note that a basis is an *ordered* set, so that two sets with the same [[vector|vectors]] but different orders form two distinct bases. A set of elements in a [[vector space]] $V$ is called a *maximal linearly independent set* if including any other vector of $V$ in the set would make it [[Linear Independence|linearly dependent]]. By definition, this means that any other vector in $V$ can be expressed as a linear combination of elements in a maximal set, so a maximal set forms a *basis* for $V$. # Basis Vectors in Differential Geometry