Vector space and subspace axioms

The axioms for fields, vector spaces, and vector subspaces are the basic operations of § Linear Algebra.

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Vector spaces §

Definition: Vector space

A vector space ( $V,+,\cdot$ ) over the field $\mathbb{F}$ consists of the following data:

• (D1) A set $V$ of vectors.
• (D2) An addition map $+:V\times V\to V$, meaning for each pair of vectors $v,w\in V$, we define a sum $v+w\in V$.
• (D3) A scalar multiplication map $\cdot :\mathbb{F}\times V\to V$, meaning for each scalar $a\in \mathbb{F}$ and each vector $v\in V$, we define a scalar multiplication $a\cdot v\in V$. In other words, the scalar multiplication function outputs a new vector in the vector space $V$.

Vector space axioms §

• (V1) Associativity. Addition and scalar multiplication in $V$ are associative.
• (V2) Commutativity. Addition is commutative.
• (V3) Distributivity. Addition distributes over scalar multiplication: for all $a\in \mathbb{F}$ and all $v,w\in V$, we have $a(v+w)=av+aw$.
• (V4) Additive identity. There exists an additive identity, the “zero vector” $\vec{0}\in V$, so that for all $v\in V$ we have $\vec{0}+v=v.$
• (V5) Additive inverse. For any $v\in V$, there exists a $w\in V$ so that $v+w=\vec{0}$. We usually denote $w$ by the symbol $-v$.
• (V6) Multiplicative identity. If $1\in \mathbb{F}$ is the multiplicative identity, then for any $v\in V$, we have $1\cdot v=v$.

Vector space lemmas §

(Proposition 22) Let $V$ be a vector space over a field $\mathbb{F}$. Then the following are true:

• Additive identities in $V$ are unique.
• Additive inverses in $V$ are unique.
• If $a\in \mathbb{F}$ and $v\in V$, then $a\cdot v=\vec{0}$ if and only if either $a=0$ or $v=\vec{0}$.
• The additive inverse to $v\in V$ is given by $-v=(-1)\cdot v$, the scalar multiplication of $v$ by $-1\in \mathbb{F}$.

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Subspaces of vector spaces §

Definition: Linear subspace

Suppose $V$ is a vector space over the field $\mathbb{F}$. A subset $W\subset V$ is said to be a linear subspace of $V$ if the following three conditions hold:

• (S1) $W$ is closed under addition. For all $w_1,w_2\in W$, we have $w_1+w_2\in W.$
• (S2) $W$ is closed under scalar multiplication. For all $c\in \mathbb{F}$ and all $w\in W$, we have $cw\in W$.
• (S3) We have $\vec{0}\in W$, so that $W$ is not the empty set $\varnothing$.

Definition: Complementary subspaces

If $V$ is a finite-dimensional vector space, the subspaces $W,U\subset V$ are complementary if both of the following are true:

• For every $v\in V$, there exist $w\in W$ and $u\in U$ so that $v=w+u$;
• $U\cap W=\{\vec{0}\}$; that is, $U,W$ intersect trivially.

Subspace propositions §

• (Proposition 23) Consider $\mathbb{F}$ as a vector space over itself. If $V\subset \mathbb{F}$ is a linear subspace, then $V=\{\vec{0}\}$ or $V=\mathbb{F}$.
• (Proposition 24) If $W\subset V$ is a linear subspace of the vector space $V$, then $W$ is once again a vector space.
  • Definitions of addition and scalar multiplication are the same operations from $V$.
  • Axioms (A1)-(A3) and (A6) follow from the corresponding axioms for $V$.
  • (A4) is true since (S3) asserts the existence of $\vec{0}$, which we can prove is the additive identity since the addition operation is the same as that of $V$.
  • (A5) is true since the existence of additive inverses in $V$ are given by $-1\cdot v$, and $W$ is closed under scalar multiplication.