 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,V2, . . . , vn} be a finite subset of a vector space V over a field F.
• (i) A linear combination of S is an expression of the form + a2V2 + ■ ■ ■ + anvn, where each 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,a2,... ,an, not all zero, such that aiv + a2v2 + ■ ■ ■ + anvn = 0. If no such scalars exist, then 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 {eb e2,... , e„}, where e, is a vector with aim the ith 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 