Engineering / Mathematics — Modules, Vector Spaces and Matrices
Vector Spaces and Dimension
Vector spaces over a field, the well-definedness of dimension, and the rank-nullity relation.
Executive summary
A vector space is a module over a field, and the invertibility of non-zero scalars makes the theory uniformly well behaved: bases always exist and all have the same size.
Dimension is the resulting invariant, and rank-nullity is the accounting identity that governs every linear computation.
Learning objectives
- State the dimension theorem.
- Apply rank-nullity.
- Compute dimensions of finite field extensions.
01Dimension is well defined
Dimension theorem
Every vector space has a basis, and any two bases have the same cardinality. That common cardinality is the dimension.
Well-definedness follows from the exchange lemma applied in both directions: each basis is independent and the other is spanning, so each has size at most the other's.
| Space | Field | Dimension |
|---|---|---|
| F^n | F | n |
| Polynomials of degree < n | F | n |
| F[X] | F | Infinite, countable |
| F_{p^k} | F_p | k |
| Matrices m × n | F | mn |
The finite field row is the one used constantly in this collection: the field with p^k elements is a k-dimensional vector space over its prime subfield, which is what makes its elements representable as coefficient vectors.
02Rank and nullity
Rank-nullity
For a linear map f: V → W with V finite dimensional,
dim V = dim(ker f) + dim(im f).
This is the first isomorphism theorem combined with the additivity of dimension across a quotient. It is the identity behind every rank computation.
- Full rank square map
nullity 0Invertible; unique solution to any system - Rank r < n
nullity n − rSolution set is a coset of an (n−r)-dimensional kernel - Zero map
nullity nKernel is everything
03Extension degrees
When a field L contains a field K, then L is a vector space over K, and its dimension is the degree of the extension.
Tower law
For fields K ⊆ L ⊆ M with finite degrees,
[M : K] = [M : L] · [L : K].
The proof is a basis construction: products of a basis for M over L with one for L over K give a basis for M over K.
For finite fields this gives the subfield structure immediately: F_{p^d} sits inside F_{p^k} exactly when d divides k, since degrees must multiply. That divisibility condition governs the distinct degree factorisation algorithm.
04Frequently asked questions
Does every vector space have a finite dimension?
No. F[X] is infinite dimensional over F, with basis the powers of X. Infinite-dimensional spaces still have bases, but the exchange argument requires Zorn's lemma.
Is rank-nullity valid over a general ring?
Not in the stated form, since dimension may be undefined. Over a principal ideal domain a rank version holds for free modules, but torsion complicates the general statement.
Why does the tower law matter for finite fields?
Because it forces subfield degrees to divide the extension degree. That is why F_{p^k} contains exactly one copy of F_{p^d} for each divisor d of k, and no other subfields.
Sources and method
Structural reference: Victor Shoup, A Computational Introduction to Number Theory and Algebra, Version 1, Cambridge University Press, 2005 — book pages 309-315.
This page carries the durable method layer only: definitions, constructions, algorithms, complexity results and selection criteria, authored originally for KEVOS. No text is transcribed or paraphrased from the source, and no numeric tables or benchmark data are reproduced — these are routed to live authoritative sources instead.
Author: Kevin Jogin. Last reviewed 2026-08-07.