The cardinal number of a basis of a vector space; the minimum number of independent coordinates required to represent any vector in that space (finite or infinite cardinal).

Principle

A vector space's dimension equals the size of any of its bases; bases are maximal linearly independent sets and minimal spanning sets, and all bases have equal cardinality when defined.

Demonstration

R^3 has dimension 3 because the standard basis {(1,0,0),(0,1,0),(0,0,1)} consists of three linearly independent vectors that span R^3. An infinite-dimensional example is the space of polynomials, which has a countably infinite basis {1,x,x^2,...}.

Misapplication

Calling the number of coordinates in a particular representation the dimension when the representation is redundant (using a nonminimal spanning set) or applying the finite-dimensional intuition to infinite-dimensional spaces (ignoring issues of Hamel bases and the Axiom of Choice).

Consequence

Dimension determines degrees of freedom, the form of linear maps (matrix sizes for finite-dimensional spaces), and many invariants (rank–nullity, classification of finite-dimensional vector spaces).

Reversal

Instead of counting independent directions, consider codimension: the number of independent linear constraints that reduce a space to a subspace; conceptually inverting degrees of freedom to constraints.

Boundary

Applies to vector spaces over a field with a well-defined notion of linear independence. Excludes other notions of 'dimension' such as topological, Hausdorff, or fractal dimensions, and in infinite cases the existence of a Hamel basis may rely on the Axiom of Choice.

Semantic Tension

Algebraic (basis) dimension can conflict with topological or measure-theoretic notions of dimension (for example, an infinite-dimensional Hilbert space vs. its topological dimension); the same word also names distinct invariants in other fields.

Synthesis

Dimension, in this algebraic sense, is the invariant cardinality of any basis of a vector space and encapsulates how many independent parameters are required to specify an arbitrary vector, subject to caveats in infinite settings.