$\mathbb{R}^n = \{(a_1, a_2, \ldots, a_n) \mid a_i \in \mathbb{R}\}$ with componentwise addition and scalar multiplication:

$(a_1, \ldots, a_n) + (b_1, \ldots, b_n) = (a_1 + b_1, \ldots, a_n + b_n), \quad c(a_1, \ldots, a_n) = (ca_1, \ldots, ca_n).$

The zero vector is $(0, 0, \ldots, 0)$. This is the prototypical vector space.

$\mathbb{C}^n$ is a vector space over $\mathbb{C}$ with the same operations as $\mathbb{R}^n$. Note that $\mathbb{C}^n$ can also be viewed as a vector space over $\mathbb{R}$ (of dimension $2n$).

The set of all $m \times n$ matrices with entries in $F$, with the usual matrix addition and scalar multiplication, forms a vector space over $F$. The zero vector is the zero matrix.

$P_n(F) = \{a_0 + a_1 x + \cdots + a_n x^n \mid a_i \in F\}$ is the space of polynomials of degree at most $n$. With the usual addition and scalar multiplication of polynomials, $P_n(F)$ is a vector space over $F$.

$P(F) = \bigcup_{n=0}^{\infty} P_n(F)$ is the space of all polynomials, which is an infinite-dimensional vector space.

Let $S$ be any nonempty set. The set $F(S, \mathbb{R}) = \{f : S \to \mathbb{R}\}$ of all real-valued functions on $S$ is a vector space with pointwise operations:

$(f + g)(s) = f(s) + g(s), \quad (cf)(s) = c \cdot f(s).$

The zero vector is the zero function $f(s) = 0$ for all $s$.

$C[a, b] = \{f : [a, b] \to \mathbb{R} \mid f \text{ is continuous}\}$ is a vector space over $\mathbb{R}$. The sum of continuous functions is continuous, and a scalar multiple of a continuous function is continuous. This is a subspace of $F([a,b], \mathbb{R})$.

$V = \{0\}$ consisting of just the zero vector is a vector space over any field $F$. It is the smallest possible vector space, with $\dim V = 0$.

Any field $F$ is a vector space over itself, with scalar multiplication being the field multiplication. More generally, if $E/F$ is a field extension, then $E$ is a vector space over $F$. For instance, $\mathbb{C}$ is a 2-dimensional vector space over $\mathbb{R}$, with basis $\{1, i\}$.

The set $F^\infty = \{(a_1, a_2, a_3, \ldots) \mid a_i \in F\}$ of all infinite sequences is a vector space over $F$ with componentwise operations. Notable subspaces include:

• $\ell^2$: sequences with $\sum |a_i|^2 < \infty$ (a Hilbert space)
• $c_0$: sequences converging to $0$
• $c_{00}$: sequences with only finitely many nonzero terms

The set of all solutions to a homogeneous linear system $Ax = 0$ (where $A$ is an $m \times n$ matrix) is a vector space over $F$: if $Au = 0$ and $Av = 0$, then $A(u + v) = 0$ and $A(cu) = 0$. This is a subspace of $F^n$.

$\mathbb{C}$ is a vector space over $\mathbb{R}$ with basis $\{1, i\}$. Every complex number $z = a + bi$ is a unique linear combination of $1$ and $i$ with real coefficients. Thus $\dim_{\mathbb{R}} \mathbb{C} = 2$.

$\mathbb{R}$ is a vector space over $\mathbb{Q}$. However, this vector space is infinite-dimensional (in fact, uncountably so). A basis is called a Hamel basis, and its existence requires the axiom of choice.