68 I The Expanding Universe of Numbers
I :V →Vdefined byIv =vfor everyv ∈Vis clearly linear. If a linear map
T:V→Wis bijective, then its inverse mapT−^1 :W→Vis again linear.
IfT:V→Wis a linear map, then the setNof allv∈Vsuch thatTv=0is
a subspace ofV, called thenullspaceorkernelofT.SinceTv=Tv′if and only if
T(v−v′)=0, the mapTis injective if and only if its kernel is{ 0 },i.e.whenTis
nonsingular.
For any subspaceUofV, its imageTU={Tv:v∈U}is a subspace ofW.In
particular,TVis a subspace ofW, called therangeofT. Thus the mapTis surjective
if and only if its range isW.
IfVis finite-dimensional, then the rangeRofTis also finite-dimensional and
dimR=dimV−dimN,
(sinceR≈V/N). The dimensions ofRandNare called respectively therankand
nullityofT. It follows that, if dimV=dimW,thenTis injective if and only if it is
surjective.
Two vector spacesV,Wover the same division ringDare said to beisomorphic
if there exists a bijective linear mapT :V →W.Asanexample,ifVis ann-
dimensional vector space over the division ringD,thenVis isomorphic toDn.Forif
v 1 ,...,vnis a basis forVand ifv=α 1 v 1 +···+αnvnis an arbitrary element ofV,
the mapv→(α 1 ,...,αn)is linear and bijective.
Thus there is essentially only one vector space of given finite dimension over a
given division ring. However, vector spaces do not always present themselves in the
concrete formDn. An example is the set of solutions of a system of homogeneous
linear equations with real coefficients. Hence, even if one is only interested in the
finite-dimensional case, it is still desirable to be acquainted with the abstract definition
of a vector space.
LetVandWbe vector spaces over the same division ringD. We can define the
sum S+Tof two linear mapsS:V→WandT:V→Wby
(S+T)v=Sv+Tv.This is again a linear map, and it is easily seen that with this definition of addition
the set of all linear maps ofVintoWis a commutative group. IfDis a field, i.e. if
multiplication inDis commutative, then for anyα∈Dthe mapαTdefined by
(αT)v=α(Tv)is again linear, and with these definitions of addition and multiplication by a scalar the
set of all linear maps ofVintoWis a vector space overD. (If the division ringDis not
a field, it is necessary to consider ‘right’ vector spaces overD,aswellas‘left’ones.)
IfV=W, then theproduct T Sis also defined and it is easily verified that the set
of all linear maps ofVinto itself is a ring, with the identity mapIas identity element
for multiplication. The bijective linear maps ofVto itself are the units of this ring and
thus form a group under multiplication, thegeneral linear group GL(V).
Similarly to the direct product of two groups and the direct sum of two rings, one
may define thetensor product V⊗V′of two vector spacesV,V′and theKronecker
product T⊗T′of two linear mapsT:V→WandT′:V′→W′.
Thecentreof a ringRis the set of allc∈Rsuch thatac=cafor everya∈R.An
associative algebra Aover a fieldFis a ring containingFin its centre. On account of