Tags:definition |space

Vector Space

Statements

Definition. A vector space over a field $F$ is a set $V$ together with two binary operations that satisfy the eight axioms listed below. In this context, the elements of $V$ are commonly called vectors, and the elements of $F$ are called scalars.

• The first operation, called vector addition or simply addition assigns to any two vectors $v_1$ and $v_1$ in $V$ a third vector in $V$ which is commonly written as $v_1+v_2$, and called the sum of these two vectors.
• The second operation, called scalar multiplication, assigns to any scalar $\alpha$ in $F$ and any vector $v$ in $V$ another vector in $V$, which is denoted $\alpha v$.

For having a vector space, the eight following axioms must be satisfied for every $v_1,v_2$ and $v_3$ in $V$, and $\alpha$ and $\beta$ in $F$.

Axiom	Meaning
Associativity of vector addition	$v_1+(v_2+v_3)=(v_1+v_2)+v_3$
Commutativity of vector addition	$v_1+v_2=v_2+v_1$
Identity element of vector addition	There exists an element $0\in V$, called the zero vector, such that $v+0=v$, for every $v\in V$
Inverse elements of vector addition	For every $v\in V$, there exists an element $-v\in V$, called the additive inverse of $v$, such that $v+(-v)=0$
Compatibility of scalar multiplication with field multiplication	For every $v\in V$, $\alpha (\beta v)=(\alpha \beta )v$
Identity element of scalar multiplication	For every $v\in V$, $1v=v$, where 1 denotes the multiplicative identity in $F$.
Distributivity of scalar multiplication with respect to vector addition	$\alpha (v_1+v_2)=\alpha v_1+\alpha v_2$
Distributivity of scalar multiplication with respect to field addition	For every $v\in V$, $(\alpha +\beta )v=\alpha v+\beta v$

References

https://en.wikipedia.org/wiki/Vector_space