# Vector spaces

- 2.199 Definition A
*vector space V*over a field F is an abelian group*(V*, +), together with a multiplication operation • : F x*V*—*> V*(usually denoted by juxtaposition) such that for all*a,b*€ F and*v, w*e*V,*the following axioms are satisfied. - (i)
*a(v + w) = av + aw.* - (ii) (a +
*b)v = av*+*bv.* - (iii) (
*ab)v = a(bv).* - (iv)
*v*=*v.*

The elements of *V* are called *vectors,* while the elements of F are called *scalars.* The group operation + is called *vector addition,* while the multiplication operation is called *scalar multiplication.*

- 2.200 Definition Let
*V*be a vector space over a field*F.*A*subspace*of*V*is an additive subgroup*U*of*V*which is closed under scalar multiplication, i.e.,*av*e*U*for all*a*e*F*and*v*e*U.* - 2.201 Fact A subspace of a vector space is also a vector space.
- 2.202 Definition Let 5 =
*{vi,V**2**,*. . . ,*v*be a finite subset of a vector space_{n}}*V*over a field*F.* - (i) A
*linear combination*of*S*is an expression of the form+ *a*_{2}V*2**+ ■ ■ ■*+*a*where each_{n}v_{n},*a,*6*F.* - (ii) The
*span*of 5, denoted (5), is the set of all linear combinations of*S.*The span of*S*is a subspace of*V.* - (iii) If
*U*is a subspace of*V,*then S is said to*span U*if*(S) = U.* - (iv) The set
*S*is*linearly dependent*over*F*if there exist scalars*a,a*not all zero, such that_{2},... ,a_{n},*aiv*+*a*0. If no such scalars exist, then_{2}v_{2}+ ■ ■ ■ + a_{n}v_{n}=*S*is*linearly independent*over*F.* - (v) A linearly independent set of vectors that spans
*V*is called a*basis*for*V.* - 2.203 Fact Let У be a vector space.
- (i) If
*V*has a finite spanning set, then it has a basis. - (ii) If
*V*has a basis, then in fact all bases have the same number of elements. - 2.204 Definition If a vector space
*V*has a basis, then the number of elements in a basis is called the*dimension*of*V,*denoted dim*V.* - 2.205 Example If
*F*is any field, then the n-fold Cartesian product*V = FxFx---xF*is a

*
*

vector space over *F* of dimension *n.* The *standard basis* for *V* is {e_{b} *e _{2},*... , e„}, where e, is a vector with aim the i

^{th}coordinate and 0’s elsewhere. □

2.206 Definition Let *E* be an extension field of *F.* Then *E* can be viewed as a vector space over the subfield *F,* where vector addition and scalar multiplication are simply the field operations of addition and multiplication in *E.* The dimension of this vector space is called the *degree* of *E* over *F,* and denoted by *[E :* Fj. If this degree is finite, then *E* is called a *finite extension* of *F.*

2.207 Fact Let *F, E,* and *L* be fields. If *L* is a finite extension of *E* and *E* is a finite extension of *F,* then *L* is also a finite extension of *F* and