What is Vector space?

Definition 18.1 University Mathematics — Year 1 · Chapter 18 — Vector Spaces

A $K$ -vector space is a set $E$ with an addition making $(E, +)$ an abelian group (zero written $0_E$ or $0$ ), and a scalar multiplication $K \times E \to E$ such that, for all $\lambda, \mu \in K$ and $x, y \in E$ :

\lambda(x + y) = \lambda x + \lambda y,\quad (\lambda + \mu) x = \lambda x + \mu x,\quad \lambda(\mu x) = (\lambda\mu) x,\quad 1\,x = x .

Consequences: $0\,x = 0_E$ , $\lambda\,0_E = 0_E$ , $(-1)x = -x$ , and $\lambda x = 0_E \implies \lambda = 0$ or $x = 0_E$ (multiply by $\lambda^{-1}$ ).

Examples

Example 18.2

$K^n$ (coordinatewise operations); the polynomials $K[X]$ ; the functions $\mathcal{F}(A, K)$ from any set $A$ to $K$ (pointwise operations) — containing continuous functions, sequences $\mathcal{F}(\N, \R)$ , etc.; $\C$ as an $\R$ -vector space. In each case the axioms are inherited from those of $K$ .

Read in context →

Examples

Related terms