Definition: A vector space over a field $F$ is a set $V$ together with two operations:
- Vector addition: $+: V \times V \to V$
- Scalar multiplication: $\cdot: F \times V \to V$
satisfying the following axioms for all $u, v, w \in V$ and $a, b \in F$:
- A1: $u + v = v + u$ (commutativity of addition)
- A2: $(u + v) + w = u + (v + w)$ (associativity of addition)
- A3: $\exists 0 \in V$ such that $v + 0 = v$ (additive identity)
- A4: $\forall v, \exists (-v)$ such that $v + (-v) = 0$ (additive inverse)
- M1: $a(bv) = (ab)v$ (associativity of scalar multiplication)
- M2: $1 \cdot v = v$ (scalar identity)
- D1: $a(u + v) = au + av$ (distributivity over vectors)
- D2: $(a + b)v = av + bv$ (distributivity over scalars)
Terminology:
- Elements of $V$ are called vectors
- Elements of $F$ are called scalars
- $F$ is typically $\mathbb{R}$ (real numbers) or $\mathbb{C}$ (complex numbers)
Parents
Children
- Definition of scalarDefinition: A scalar is a single number from a field \(\mathbb{F}\), typically \(\mathbb{R}\) (real numbers) or...
- Definition of vectorA vector is an element of a specified vector space. In $\mathbb F^n$, vectors are ordered $n$-tuples of scalars,...
- Definition of vector-scalar multiplicationDefinition: The scalar multiplication of a vector \(\mathbf{v} = (v 1, v 2, \ldots, v n) \in \mathbb{F}^n\) by a...
- Axioms of a vector spaceDefinition: A vector space over a field \(\mathbb{F}\) is a set \(V\) together with two operations --- vector...
- Definition of vector additionVector addition is the operation $(u,v)\mapsto u+v$ in a vector space. It is associative and commutative, has a zero...
- The additive inverse of a vector is called the negative of the vector.The negative of $v$, denoted $-v$, is its additive inverse, satisfying $v+(-v)=0$. It equals $(-1)v$. In coordinates,...
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