Consider the following problem. You want to describe the set of all solutions to the linear system

\begin{align*} 2x_1 - x_2 + 3x_3 &= 0 \\ x_1 + x_2 - x_3 &= 0. \end{align*}

You find one solution $v = (1, 1, 0)$ and another $w = (1, -1, 1)$. You notice immediately that $v + w = (2, 0, 1)$ is also a solution, and that $3v = (3, 3, 0)$ is also a solution. You could verify each by substituting back, but you also sense that something deeper is at work: the set of solutions is *closed under addition and scalar multiplication*, and this closure is not an accident — it is a consequence of linearity. The set of solutions forms a structure that mathematicians have isolated and named, because it appears everywhere: in systems of differential equations, in quantum mechanics, in signal processing, in geometry. That structure is a **vector space**.

The power of abstraction here is immense. Once you know you are working inside a vector space, an enormous library of results becomes available to you — about dimension, about linear maps, about decompositions, about geometry — without ever returning to the particular system that generated the space. The point of the definition is not to describe one example but to capture the common skeleton shared by $\mathbb{R}^n$, spaces of polynomials, spaces of continuous functions, spaces of matrices, and countless others.

[example: Solution Sets Are Closed Under Linear Combinations] Let $A \in \mathbb{R}^{m \times n}$ and consider the homogeneous system $Ax = 0$. If $v, w \in \mathbb{R}^n$ satisfy $Av = 0$ and $Aw = 0$, then for any scalars $\alpha, \beta \in \mathbb{R}$, \begin{align*} A(\alpha v + \beta w) &= \alpha Av + \beta Aw = \alpha \cdot 0 + \beta \cdot 0 = 0. \end{align*} So $\alpha v + \beta w$ is also a solution. This computation uses only two properties of matrix multiplication: it distributes over addition, and it commutes with scalar multiplication. These are precisely the axioms that define a vector space. The set of solutions $\ker(A) := \{x \in \mathbb{R}^n : Ax = 0\}$ is not just a set — it is a vector space in its own right. [/example]

## Definition

To isolate what matters — what makes the solution-space example work, and what makes $\mathbb{R}^n$, polynomials, and continuous functions all behave the same way — we abstract away everything except the two operations and the rules they must obey.

[definition: Vector Space] Let $F$ be a field. A **vector space** over $F$ is a set $V$ together with two operations, \begin{align*} +: V \times V &\to V & &\text{(vector addition)} \\ \cdot: F \times V &\to V & &\text{(scalar multiplication),} \end{align*} satisfying the following eight axioms. For all $u, v, w \in V$ and all $\alpha, \beta \in F$: 1. **(A1) Associativity of addition:** $(u + v) + w = u + (v + w)$. 2. **(A2) Commutativity of addition:** $u + v = v + u$. 3. **(A3) Additive identity:** There exists $0 \in V$ such that $v + 0 = v$ for all $v \in V$. 4. **(A4) Additive inverses:** For each $v \in V$ there exists $-v \in V$ with $v + (-v) = 0$. 5. **(S1) Compatibility:** $\alpha(\beta v) = (\alpha\beta)v$. 6. **(S2) Identity scalar:** $1 \cdot v = v$. 7. **(D1) Distributivity over vector addition:** $\alpha(u + v) = \alpha u + \alpha v$. 8. **(D2) Distributivity over scalar addition:** $(\alpha + \beta)v = \alpha v + \beta v$. Elements of $V$ are called **vectors**; elements of $F$ are called **scalars**. [/definition]

The word "field" here means a set equipped with addition and multiplication satisfying the usual rules — think $\mathbb{R}$, $\mathbb{C}$, $\mathbb{Q}$, or $\mathbb{F}_p$ for a prime $p$. In most applications in linear algebra the field is $\mathbb{R}$ or $\mathbb{C}$, and we speak of a **real vector space** or **complex vector space** respectively.

It is worth pausing on what this definition does *not* say. There is no mention of coordinates, of arrows, of length, or of angles. Addition and scalar multiplication are abstract operations whose output is required only to satisfy eight equations. This generality is the whole point: it lets us treat spaces of functions, spaces of matrices, and $\mathbb{R}^n$ all by the same machinery.

[remark: Uniqueness of the Zero Vector and Additive Inverses] The axioms guarantee existence but not uniqueness of the zero vector and additive inverses. Uniqueness follows at once: if $0$ and $0'$ both satisfy (A3), then $0 = 0 + 0' = 0'$. Similarly, if both $-v$ and $-'v$ are additive inverses of $v$, then \begin{align*} -v &= -v + 0 = -v + (v + (-'v)) = (-v + v) + (-'v) = 0 + (-'v) = -'v. \end{align*} These are not separate axioms to impose — they are theorems that come for free. [/remark]

[example: Continuous Functions on an Interval] The space $C([0, 1])$ of continuous real-valued functions on $[0, 1]$, with pointwise addition and scalar multiplication \begin{align*} (f + g)(t) &:= f(t) + g(t), \qquad (\alpha f)(t) := \alpha f(t), \end{align*} is a real vector space. Checking one non-obvious axiom: the zero vector is the constant function $\mathbf{0}(t) = 0$, and for each $f \in C([0,1])$ the additive inverse is $(-f)(t) = -f(t)$, which is continuous whenever $f$ is. All eight axioms are inherited from the pointwise arithmetic of $\mathbb{R}$. What makes this example worthwhile is that here the "vectors" are functions, objects with infinitely many degrees of freedom. The dimension of $C([0,1])$ is infinite — we will see why when we discuss bases — yet every element obeys exactly the same algebraic rules as an $n$-tuple. The same abstract framework covers both. More concretely: the differential equation $f'' + f = 0$ has the solution set $\{A\sin t + B\cos t : A, B \in \mathbb{R}\}$, a two-dimensional subspace of $C([0, 2\pi])$. The fact that this set of solutions is a vector space is not a coincidence — it is a consequence of the linearity of differentiation, and it is precisely the kind of structure the vector space axioms are designed to capture. [/example]

[example: Spaces of Functions] Let $X$ be any nonempty set and $F$ a field. Define $F^X$ to be the set of all functions $f: X \to F$. With pointwise operations, \begin{align*} (f + g)(x) &:= f(x) + g(x), \qquad (\alpha f)(x) := \alpha f(x), \end{align*} $F^X$ is a vector space over $F$. The zero vector is the constant function $0: x \mapsto 0_F$. Each $f$ has additive inverse $(-f): x \mapsto -f(x)$. All eight axioms follow from the corresponding axioms in $F$. When $X = \{1, 2, \ldots, n\}$, we recover $F^n$: a function $f: \{1, \ldots, n\} \to F$ is exactly an $n$-tuple $(f(1), f(2), \ldots, f(n))$. When $X = [0, 1]$ and $F = \mathbb{R}$, this gives the space of all real-valued functions on $[0, 1]$, which contains $C([0, 1])$, $C^\infty([0, 1])$, $L^2([0, 1])$, and many other important spaces as subsets (and, under appropriate conditions, as subspaces). [/example]

[example: Polynomial Spaces] Fix a field $F$ and let $\mathcal{P}(F)$ denote the set of all polynomials with coefficients in $F$, \begin{align*} \mathcal{P}(F) &= \left\{ a_0 + a_1 t + a_2 t^2 + \cdots + a_n t^n : n \ge 0,\, a_i \in F \right\}. \end{align*} Addition and scalar multiplication are the familiar polynomial operations. This is a vector space. The zero vector is the zero polynomial. For each $n \ge 0$, the subset $\mathcal{P}_n(F) \subset \mathcal{P}(F)$ of polynomials of degree at most $n$ is also a vector space. The space $\mathcal{P}_n(F)$ has dimension $n+1$ — something we will make precise shortly. [/example]

## Subspaces

Not every subset of a vector space is again a vector space, but some are, and they inherit all the structure from the ambient space. Identifying subspaces is often the first step in understanding the geometry of a space.

The naive criterion for checking that a subset $W \subset V$ is a vector space in its own right would be to verify all eight axioms. But axioms (A1), (A2), (S1), (S2), (D1), (D2) are inherited automatically from $V$ — they hold for all vectors in $V$, so in particular they hold for vectors in $W$. What can fail is closure (the addition or scalar multiplication of two vectors in $W$ might escape $W$), existence of a zero vector, and existence of additive inverses. The subspace criterion packages exactly the minimal checks needed.

[definition: Subspace] Let $V$ be a vector space over a field $F$. A nonempty subset $W \subset V$ is a **subspace** of $V$ if it is closed under addition and scalar multiplication: \begin{align*} u, v \in W,\, \alpha \in F \implies u + v \in W \text{ and } \alpha u \in W. \end{align*} [/definition]

Equivalently, $W$ is a subspace if for all $u, v \in W$ and $\alpha, \beta \in F$,

\begin{align*} \alpha u + \beta v \in W. \end{align*}