All you do is referring to the following definition:
Definition: A<span> vector space </span>is a set V on which two operations + and · are defined,
called<span> vector
addition </span>and<span> scalar multiplication.</span>
The operation + (vector addition) must satisfy the
following conditions:
Closure: If u and v are any vectors in
V, then the sum u + v belongs to V.
(1)<span> Commutative law: </span>For all vectors u and v in V, u + v = v + u
(2)<span> Associative law: </span>For all vectors u, v, w in V, u <span>+ (v</span> + w<span>) = (u</span> + v) + w
(3)<span> Additive identity: </span>The set V contains an<span> additive identity </span>element, denoted by 0, such that for
any vector v in V, 0 + v = v and v + 0 = v.
(4)<span> Additive inverses: </span>For each vector v in V, the equations v + x = 0 and x + v = 0 have
a solution x in V, called an<span> additive inverse </span>of v, and denoted by - v.
The operation · (scalar
multiplication) is defined between real numbers (or scalars) and vectors, and
must satisfy the following conditions:
Closure: If v in any vector in
V, and c is any real number, then the product c · v belongs to
V.
(5)<span> Distributive law: </span>For all real numbers c and all vectors u, v in V, c · <span>(u</span> + v) = c · u + c · v
(6)<span> Distributive law</span>: For all real
numbers c, d and all vectors v in V, (c+d) · v = c · v + d · v
(7)<span> Associative law</span>: For all real
numbers c,d and all vectors v in V, c · (d · v) = (cd) · v
(8)<span> Unitary law</span>: For all vectors v in V,
1 · v = <span>v</span>
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