Basis and Dimension

The Coordinate Skeleton

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

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

A basis is thus a minimal spanning set — just enough vectors, with none wasted, to build everything in $V$. Removing any basis vector loses spanning; adding any vector loses independence. The standard basis of $\mathbb{R}^n$ is $e_1 = (1, 0, \dots), e_2 = (0, 1, 0, \dots), \dots$, but every space has infinitely many bases.

The single most important fact: every basis of a given space has the same number of vectors. That number is the dimension of the space, $\dim V$. The plane has dimension 2, ordinary space dimension 3, the space of polynomials of degree $\le n$ has dimension $n+1$. Dimension is a fundamental invariant — it does not depend on which basis you pick.

Bases give coordinates: because a basis spans, every vector is some combination of the basis vectors; because it is independent, that combination is unique. So each vector corresponds to exactly one list of coefficients — its coordinates in that basis. This is precisely why coordinates work, and it is the bridge from abstract vector spaces back to concrete tuples of numbers.

Two convenient consequences in an $n$-dimensional space: any $n$ independent vectors automatically form a basis (independence plus the right count forces spanning), and any $n$ spanning vectors are automatically independent. So once you know the dimension, you only need to check one of the two basis conditions.

Common pitfall: thinking any spanning set is a basis, or any independent set is a basis. A basis needs both properties. A spanning set with extra, dependent vectors is not a basis (too many); an independent set too small to reach everything is not a basis (too few). And every basis of the space has exactly $\dim V$ vectors — no more, no fewer.