Dimension

Precise definition

The dimension of a finite-dimensional vector space $V$, written $\dim V$, is the number of vectors in any basis of $V$. The basis theorem guarantees this number is well-defined. The zero space has dimension 0 because its basis is the empty list.

Notation and mathematical language

Examples: $\dim\mathbb R^n=n$; polynomials of degree at most $m$ have basis $1,x,\ldots,x^m$ and dimension $m+1$; $m\times n$ matrices have dimension $mn$. For $T:V\to W$, rank–nullity says $\dim V=\dim\ker T+\dim\operatorname{range}T$ when $V$ is finite-dimensional.

Conceptual picture

Dimension counts independent directions, not the number of vectors in the space. A plane through the origin contains infinitely many vectors but needs two basis directions. Constraints reduce dimension when they are independent.

Conditions and key results

The subset must first be a vector space before dimension is assigned in this sense. A list longer than $\dim V$ is dependent; a list shorter than $\dim V$ cannot span. For solution spaces, rank and number of free variables must be computed over the stated field.

A reliable strategy

1. Verify the object is a vector space or subspace under the stated operations.
2. Find a spanning description and remove dependent vectors, or solve constraints using free parameters.
3. Exhibit a basis and count its elements; use rank–nullity when a linear map is available.
4. Check the count against ambient dimension and the number of independent constraints.

Fully worked example

Interpretation and application

Dimension predicts degrees of freedom in models, data subspaces, polynomial approximation, and solution sets. It is invariant under a change of basis, so different coordinate descriptions do not change the underlying count.

Common mistakes

Verification and reasonableness

• Prove the proposed basis spans and is independent.
• Use rank–nullity as an independent count when the subspace is a kernel.
• Compare with the maximum possible ambient dimension.

Practice

1. What is $\dim P_4$?
2. What is the dimension of the zero subspace?
3. A $3\times5$ matrix has rank 3. What is its nullity?

Further deduction

For a homogeneous system $A\mathbf x=0$ with $n$ variables, the solution space has dimension $n-\operatorname{rank}A$. Each pivot variable is determined by free variables, but dimension counts free parameters rather than simply equations; dependent equations do not reduce the dimension again. This is the matrix form of rank–nullity.

Dimension arguments can prove impossibility without computation. Three vectors cannot span $\mathbb R^4$; five vectors cannot be independent there. A linear map $\mathbb R^5\to\mathbb R^3$ has nullity at least 2, so information loss is unavoidable. These conclusions follow before any matrix entries are known.

For finite-dimensional subspaces, $\dim(U+W)=\dim U+\dim W-\dim(U\cap W)$. The intersection correction prevents double-counting shared directions and yields immediate bounds on how large a sum can be, often avoiding a full basis computation.

Related topics

• Basis
• Subspaces
• Kernel and Range