Number Theory: An Introduction to Mathematics

9 Vector Spaces and Associative Algebras 65

As another example, the setC(I)of all continuous functionsf :I→R,where

Iis an interval of the real line, is a vector space over the fieldRof real numbers if

addition and multiplication by a scalar are defined, for everyt∈I,by

(f+g)(t)=f(t)+g(t),

(αf)(t)=αf(t).

LetVbe an arbitrary vector space over a division ringD.IfOis the identity

element ofVwith respect to addition, then

αO=O for everyα∈D,

sinceαO=α(O+O)=αO+αO. Similarly, if 0 is the identity element ofDwith

respect to addition, then

0 v=O for everyv∈V,

since 0v=( 0 + 0 )v= 0 v+ 0 v.Furthermore,

(−α)v=−(αv) for allα∈Dandv∈V,

sinceO= 0 v=(α+(−α))v=αv+(−α)v,and

αv=O ifα=0andv=O,

sinceα−^1 (αv)=(α−^1 α)v= 1 v=v.
From now on we will denote the zero elements ofDandVby the same symbol 0.
This is easier on the eye and in practice is not confusing.
A subsetUof a vector spaceVis said to be asubspaceofVif it is a vector space
under the same operations asVitself. It is easily seen that a nonempty subsetUis a
subspace ofVif (and only if) it is closed under addition and multiplication by a scalar.
For then, ifu∈U,also−u=(− 1 )u∈U,andsoUis an additive subgroup ofV.
The other requirements for a vector space are simply inherited fromV.
Forexample,if1≤m<n,thesetofall(α 1 ,...,αn)∈Dnwithα 1 = ··· =
αm =0 is a subspace ofDn. Also, the setC^1 (I)of all continuously differentiable
functionsf :I →Ris a subspace ofC(I). Two obvious subspaces of any vector
spaceVareVitself and the subset{ 0 }which contains only the zero vector.
IfU 1 andU 2 are subspaces of a vector spaceV, then theirintersection U 1 ∩U 2 ,
which necessarily contains 0, is again a subspace ofV.Thesum U 1 +U 2 , consisting
of all vectorsu 1 +u 2 withu 1 ∈U 1 andu 2 ∈U 2 , is also a subspace ofV. Evidently
U 1 +U 2 containsU 1 andU 2 and is contained in every subspace ofVwhich contains
bothU 1 andU 2 .IfU 1 ∩U 2 ={ 0 },thesumU 1 +U 2 is said to bedirect, and is denoted
byU 1 ⊕U 2 , since it may be identified with the set of all ordered pairs(u 1 ,u 2 ),where
u 1 ∈U 1 andu 2 ∈U 2.
LetVbe an arbitrary vector space over a division ringDand let{v 1 ,...,vm}be a
finite subset ofV. A vectorvinVis said to be alinear combinationofv 1 ,...,vmif

v=α 1 v 1 +···+αmvm