Vector Space

$(V,*,\circ)\;\text{over}\;(F,+,\cdot)$ is a vector space, iff it satisfies:

• $(V,*)$ is an abelian group
• $(F,+,\cdot)$ is a field
• $\forall a \in F, \forall x \in V, a \circ x \in V$
• $\forall a \in F, \forall x,y \in V, a \circ (x * y) = (a \circ x) * (a \circ y)$
• $\forall a,b \in F, \forall x \in V, (a+b) \circ x = (a \circ x) * (b \circ x)$
• $\forall a,b \in F, \forall x \in V, (a\cdot b) \circ x = a \circ (b \circ x)$

In the above definition:

• $F$ is the set of scalars
• $V$ is the set of vectors
• $+$ is number addition
• $\cdot$ is number multiplication
• $*: V \times V \to V$ is vector addition
• $\circ: F \times V \to V$ is scalar multiplication

Alternate Notation

When no confusion arises, instead of 4 different symbols:

• $+$ for both vector addition and scalar addition
• $\cdot$ for both scalar multiplication and field multiplication

And:

• Zero vector is denoted as $\underline{0}$
• $(V,*,\circ)\;\text{over}\;(F,+,\cdot)$ is simply denoted as $V$ over $F$.

Properties

Suppose $(V,*,\circ)\;\text{over}\;(F,+,\cdot)$ is a vector space.

Zero vector exists

Denoted as $e$.

Negative x is the additive inverse