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The dimension of a vector space V, denoted dim(V), is the size of any basis. All bases have equal cardinality. Dimension measures the number of independent coordinates needed. For Rn, dimension is n. Subspaces have dimension ≤ the parent space's dimension.
You learned from the definition of a basis that a basis is a set of vectors that is both linearly independent and spans the space — it is exactly enough to describe every vector in the space without redundancy. Dimension is what you get when you count the vectors in a basis. It measures how many truly independent directions exist in the space.
The foundational theorem behind dimension is that all bases of a vector space have the same number of vectors. This is not obvious — a space might seem to admit different bases of different sizes — but the exchange lemma guarantees it can't happen. If B₁ and B₂ are both bases of V, then |B₁| = |B₂|. Because of this, dim(V) is well-defined: you can compute it from any basis and get the same answer. For ℝⁿ, the standard basis {e₁, e₂, ..., eₙ} has n vectors, so dim(ℝⁿ) = n. For the space of 2×2 matrices, the standard basis has 4 matrices, so the dimension is 4, even though the space looks different from ℝ⁴.
Think of dimension as the minimum number of numbers needed to uniquely specify any element of the space. In ℝ³, you need exactly 3 coordinates — no more, no less — to pin down a point. A plane through the origin in ℝ³ is a 2-dimensional subspace: you need only 2 coordinates relative to a basis for the plane. A line through the origin is 1-dimensional. The zero vector alone forms the zero subspace, which has no basis and dimension 0. Each of these subspaces requires fewer independent coordinates than the parent space — which is why dimension of a subspace is always ≤ dimension of the full space.
Dimension interacts with the four fundamental subspaces of a matrix in a precise way. The rank of a matrix A is the dimension of its column space (equivalently, its row space). The nullity is the dimension of the null space. The rank-nullity theorem — which you'll prove next — states that rank + nullity = n, where n is the number of columns of A. This is a conservation law for dimensions: the dimensions of the column space and null space always partition the n input dimensions. Understanding dimension as a count of independent directions makes this theorem intuitive rather than mysterious.
The concept scales far beyond ℝⁿ. The space of polynomials of degree ≤ 3 is 4-dimensional (basis: {1, x, x², x³}). The space of continuous functions on [0, 1] is infinite-dimensional — no finite set of functions spans it. Dimension is the single number that classifies a finite-dimensional vector space up to isomorphism: any two vector spaces over the same field with the same dimension are structurally identical. This makes dimension one of the most powerful invariants in all of linear algebra.