# Introduction

Linear algebra is the study of vector spaces and the [linear maps](/page/Linear%20Map) between them. It is, in a sense, the simplest and most complete branch of algebra: the objects are tame enough to admit a full classification (every finite-dimensional vector space is determined by a single number, its dimension), yet rich enough to underpin vast swathes of mathematics, physics, and computation.

The subject has a dual character. On one hand, it is a *concrete* toolkit: systems of linear equations, matrix arithmetic, Gaussian elimination, eigenvalue computations. These tools are indispensable in applications ranging from data science and engineering to quantum mechanics and economics. On the other hand, linear algebra is an *abstract* structural theory: vector spaces, linear maps, duality, and spectral decomposition form a language that organises and clarifies ideas across all of mathematics. The interplay between the concrete and the abstract — between matrices and the maps they represent — is the central theme of this course.

The development begins with the axioms of a vector space and the fundamental notions of linear independence, span, and dimension. We then study linear maps, their kernels and images, and the rank-nullity theorem — the first major structural result of the theory. Next, the dual space provides a second perspective on every vector space, revealing a hidden symmetry that connects subspaces to systems of linear equations. Bilinear forms extend linearity to [functions](/page/Function) of two variables, leading naturally to determinants (which detect invertibility and measure volume distortion) and to the classification of symmetric and Hermitian forms via Sylvester's law of inertia. The theory of endomorphisms — linear maps from a space to itself — culminates in the [Jordan normal form](/theorems/864), a complete classification up to similarity over algebraically closed fields. Finally, inner product spaces specialise the theory to the positive-definite case, where algebra meets geometry: orthogonality, projection, and the spectral theorem for self-adjoint and unitary maps.

Throughout, we work over an arbitrary field $\mathbb{F}$ unless a specific choice (typically $\mathbb{R}$ or $\mathbb{C}$) is required. The treatment is rigorous and proof-driven, with each result motivated by the questions it answers and the tools it provides for later developments.

# Vector Spaces

When studying systems of linear equations, a striking pattern emerges: the set of all solutions to a homogeneous system is closed under addition and scalar multiplication. If $u$ and $v$ both solve $Ax = \mathbf{0}$, then so does $\lambda u + \mu v$ for any scalars $\lambda, \mu$. This algebraic closure is not special to solution [sets](/page/Set) of matrices — it appears equally in the space of all continuous functions on an interval, the space of polynomials of bounded degree, and the space of solutions to a linear differential equation. The goal of this chapter is to isolate the abstract structure shared by all these examples, develop the language of linear independence and bases, and prove that every finite-dimensional vector space is completely determined (up to isomorphism) by a single number: its dimension.

Throughout, $\mathbb{F}$ denotes an arbitrary field (such as $\mathbb{R}$, $\mathbb{C}$, or a finite field $\mathbb{F}_p$). All vector spaces are defined over this fixed field unless otherwise specified.

## Core Definitions

### The Vector Space Axioms

The examples above — solution sets, function spaces, polynomial spaces — all share the same algebraic operations: we can add elements and multiply by scalars, and these operations obey familiar rules (commutativity, associativity, distributivity). Rather than verify these properties separately in each context, we axiomatise them once and develop the theory in full generality. This is the power of abstraction: any result proved for an abstract vector space applies simultaneously to every concrete instance.

[definition:Vector Space] Let $\mathbb{F}$ be a field. An $\mathbb{F}$-vector space is an abelian group $(V, +)$ equipped with a scalar multiplication map \begin{align*} \mathbb{F} \times V &\to V \\ (\lambda, v) &\mapsto \lambda v, \end{align*} satisfying the following axioms for all $\lambda, \mu \in \mathbb{F}$ and all $u, v \in V$: 1. $\lambda(\mu v) = (\lambda \mu) v$ (associativity of scalar multiplication) 2. $\lambda(u + v) = \lambda u + \lambda v$ (distributivity over vector addition) 3. $(\lambda + \mu)v = \lambda v + \mu v$ (distributivity over field addition) 4. $1_{\mathbb{F}} v = v$ (identity element of scalar multiplication) [/definition]

Two points deserve emphasis. First, the definition does not require $V$ to be finite-dimensional or to carry any notion of distance, angle, or convergence — those are additional structures (norms, inner products) that we will introduce much later. Second, the requirement that $(V, +)$ is an abelian group already encodes the existence of a zero vector $\mathbf{0}$ and additive inverses $-v$ for every $v \in V$; there is no need to include these as separate axioms.

Having stated the axioms, a natural question is whether familiar arithmetic properties — such as $0 \cdot v = \mathbf{0}$ — follow from them or need to be assumed separately. The following result confirms that they are consequences, not additional assumptions.

[quotetheorem:369]

This result is more than a sanity check. It confirms that the vector space axioms are self-consistent and sufficiently strong: the familiar arithmetic of vectors is not an extra assumption but a logical consequence. Part (i) is used implicitly throughout linear algebra whenever we need to recognise a zero vector from a scalar multiplication. Part (ii) ensures that the group-theoretic additive inverse and the scalar-multiplication inverse agree, so notation like $u - v$ is unambiguous.

[citeproof:369]

### Subspaces

Given a vector space $V$, we often want to work with "smaller" vector spaces sitting inside $V$. For instance, the set of solutions to a single linear equation $a_1 x_1 + \cdots + a_n x_n = 0$ is a subset of $\mathbb{F}^n$ that is itself closed under the vector space operations. Rather than checking all four vector space axioms from scratch each time, the following definition provides a more efficient criterion: we need only check closure under addition and scalar multiplication, plus the presence of the zero vector.

[definition:Subspace] Let $V$ be an $\mathbb{F}$-vector space. A subset $U \subseteq V$ is a subspace of $V$ if: 1. $\mathbf{0} \in U$ 2. $u, v \in U$ implies $u + v \in U$ (closure under addition) 3. $u \in U$, $\lambda \in \mathbb{F}$ implies $\lambda u \in U$ (closure under scalar multiplication) [/definition]