Number Theory: An Introduction to Mathematics

9 Vector Spaces and Associative Algebras 67

IfVhas a basis containingnelements, we sayVhasdimension nand we write
dimV=n. We say thatVhas infinite dimension if it is not finitely generated, and has
dimension 0 if it contains only the vector 0.
For example, the fieldCof complex numbers may be regarded as a 2-dimensional
vector space over the fieldRof real numbers, with basis{ 1 ,i}.
Again,Dnhas dimensionnas a vector space over the division ringD, since it has
the basis

e 1 =( 1 , 0 ,..., 0 ), e 2 =( 0 , 1 ,..., 0 ),..., en=( 0 , 0 ,..., 1 ).

On the other hand, the real vector spaceC(I)of all continuous functionsf:I→R
has infinite dimension if the intervalI contains more than one point since, for any
positive integern, the real polynomials of degree less thannform ann-dimensional
subspace.
The first of these examples is readily generalized. If EandF are fields with
F ⊆E, we can regardEas a vector space overF. If this vector space is finite-
dimensional, we say thatEis afinite extensionofFand define thedegreeofEover
Fto be the dimension [E:F] of this vector space.
Any subspaceUof a finite-dimensional vector spaceVis again finite-dimensional.
Moreover, dimU≤dimV, with equality only ifU=V.IfU 1 andU 2 are subspaces
ofV,then

dim(U 1 +U 2 )+dim(U 1 ∩U 2 )=dimU 1 +dimU 2.

LetVandWbe vector spaces over the same division ringD.AmapT:V→W
is said to belinear,oralinear transformation, or a ‘vector space homomorphism’, if
for allv,v′∈Vand everyα∈D,

T(v+v′)=Tv+Tv′, T(αv)=α(Tv).

Since the first condition implies thatTis a homomorphism of the additive group ofV
into the additive group ofW, it follows thatT 0 =0andT(−v)=−Tv.
For example, if (τjk)isanm×nmatrix with entries from the division ringD,then
the mapT:Dm→Dndefined by

T(α 1 ,...,αm)=(β 1 ,...,βn),

where

βk=α 1 τ 1 k+···+αmτmk ( 1 ≤k≤n),

is linear. It is easily seen that every linear map ofDmintoDnmay be obtained in this
way.
As another example, ifC^1 (I)is the real vector space of all continuously differen-
tiable functionsf:I →R, then the mapT:C^1 (I)→C(I)defined byTf=f′
(the derivative off) is linear.
LetU,V,Wbe vector spaces over the same division ringD.IfT:V→Wand
S:U →V are linear maps, then the composite mapT◦S :U →W is again
linear. For linear maps it is customary to writeTSinstead ofT◦S. The identity map