Page 23 - Linear Algebra Done Right

Definition of Vector Space
The motivation for the definition of a vector space comes from the
important properties possessed by addition and scalar multiplication
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on F . Specifically, addition on F is commutative and associative and 9
has an identity, namely, 0. Every element has an additive inverse. Scalar
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multiplication on F is associative, and scalar multiplication by 1 acts
as a multiplicative identity should. Finally, addition and scalar multi-
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plication on F are connected by distributive properties.
We will define a vector space to be a set V along with an addition
and a scalar multiplication on V that satisfy the properties discussed
in the previous paragraph. By an addition on V we mean a function
that assigns an element u + v ∈ V to each pair of elements u, v ∈ V.
By a scalar multiplication on V we mean a function that assigns an
element av ∈ V to each a ∈ F and each v ∈ V.
Now we are ready to give the formal definition of a vector space.
A vector space is a set V along with an addition on V and a scalar
multiplication on V such that the following properties hold:

commutativity
u + v = v + u for all u, v ∈ V;

associativity
(u+v)+w = u+(v +w) and (ab)v = a(bv) for all u, v, w ∈ V
and all a, b ∈ F;

additive identity
there exists an element 0 ∈ V such that v + 0 = v for all v ∈ V;
additive inverse
for every v ∈ V, there exists w ∈ V such that v + w = 0;
multiplicative identity
1v = v for all v ∈ V;
distributive properties
a(u + v) = au + av and (a + b)u = au + bu for all a, b ∈ F and
all u, v ∈ V.
The scalar multiplication in a vector space depends upon F. Thus
when we need to be precise, we will say that V is a vector space over F
instead of saying simply that V is a vector space. For example, R n is
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a vector space over R, and C is a vector space over C. Frequently, a
vector space over R is called a real vector space and a vector space over