Lecture 7: Fields and Vector Spaces

Lecture 7: Fields and Vector Spaces 7 Fields and Vector Spaces 7.1 Review Last time, we learned that we can quotient out a normal subgroup of N to make a new group, G/N. 7.2 Fields Now, we will do a hard pivot to learning linear algebra, and then later we will begin to merge it with group theory in diferent ways. In order to defne a vector space, the underlying feld must be specifed. Defnition 7.1 A feld F is a set with the operations (+, ×). It must satisfy that • (F, +) is an abelian group with the usual rules, and • (F × := F \ {0}, ×) is an abelian group. Also, addition and multiplication must distribute over each other.a a There is some compatibility required. In essence, a feld is a set with additive and multiplicative group structures that interact in nice ways. Example 7.2 The sets C, R, and Q are felds, but not Z, since it is not invertible under multiplication. Since division does not exist in Z, it is not a feld. In fact, Q is essentially obtained from Z by making it into a feld by adding division. Example 7.2 gives us examples of felds with infnitely many elements, but felds can also be constructed that have fnite order. Indeed, there is one for every prime number p. Example 7.3 (Fields of prime order) For a prime p, (Fp = Zp , +, ×) is a feld. If a = ̸ 0 mod p, then gcd(a, p) = 1 implies that ar + ps = 1, and so ar ≡ 1 mod p, and thus a is invertible with multiplicative inverse r−1 . However, Z6 is not a feld; for example, 2 mod 6 has no inverse. In general, Zn where n is not a prime is not a feld, because there will exist some element that is not relatively prime to n, and it will not be invertible. 7.3 Vector Spaces A vector space, which may be a familiar concept from learning about matrices, can be defned over any feld. Defnition 7.4 A vector space V over a feld F is a set V with some operation + such that (V, +) is an abelian group. • We must be able to scale vectors: F ×V → V (a, ⃗v ) 7→ a⃗v . • Addition and multiplication play nicely and satisfy the usual rules (· · · , a(b⃗v ) = (ab) · ⃗v , · · · ). Lecture 7: Fields and Vector Spaces Example 7.5 For a feld F, F n , column vectors with n components (a1 , · · · , an )t , form a vector space of dimension n. Example 7.6 If A is an m×n matrix, then {⃗v ∈ F n : A⃗v = (0, · · · , 0)} is a vector space. Example 7.7 For a homogeneous linear ODE, the solutions form a vector space. 7.4 Bases and Dimension A basis of a vector space is a set of vectors providing a way of describing it without having to list every vector in the vector space. Defnition 7.8 Given ⃗v1 , ⃗v2 , · · · , ⃗vn ∈ V, a linear combination is ⃗v = X ai⃗vi for ai ∈ F. Defnition 7.9 For S = {⃗v1 , ⃗v2 , · · · , ⃗vn }, the span Span(S) = {⃗v ∈ V : ⃗v = X ai⃗vi } This is similar to generating subgroups using elements of a group G, except using the operations of vector spaces. Artin likes to use the (nonstandard) notation � ⃗v1 ···   a1 X
 .  ⃗vn  ..  := ai⃗vi an for a linear combination. Defnition 7.10 A set of vectors S spans V if Span(S) = V. a There a is at least one way of writing ⃗v as a linear combination. Defnition 7.11 A set of vectors {⃗vi } is linearly independent if X ai⃗vi = ⃗0 if and only if ai = 0 for all i.a a There is at most one way of writing ⃗v as a linear combination. A basis is both linearly independent and spans. Lecture 7: Fields and Vector Spaces Defnition 7.12 A set of vectors S = { #v»1 , · · · , ⃗vn } is a basis if S spans V and is linearly independent. Equivalently, each ⃗v ∈ V can be written uniquely as ⃗v = a1⃗v1 + · · · + an⃗vn , where the ai are called the coordinates of ⃗v in the basis S. The standard basis for R2 is     1 0 , . 0 1 In general, when we write a vector (a, b)t , it represents the linear combination a(1, 0)t + b(0, 1)t . Example 7.13 Let V = R2 . Then the set       1 3 2 , , 1 2 1 # » spans R2 , but is linearly dependent: #» v 1 − #v»2 + #» v 3 = 0 . But { #v»1 , #» v 2 } forms a basis. S= A good choice of basis often makes problems easier. Defnition 7.14 A vector space V is fnite-dimensional if V = Span({ #v»1 , · · · , #» v n }) for some #» v i ∈ V.a a Infnite-dimensional vector spaces are super interesting, but not studied in this class. Real analysis can be used to study them! Lemma 7.15 #» , · · · , w #» } is linearly independent, then If S = { #v»1 , · · · , #» v r } spans V, and L = {w 1 s 1. Removing elements of S gets a basis of V. 2. Adding elements of S to L gets another basis of V. 3. |S| ≥ |L|. Corollary 7.16 If S and L are both bases for V, then |S| = |L|. Any two bases of V contain the same number of vectors. Proof. Applying the lemma twice for S and L gives |S| ≥ |L| and |L| ≥ |S|. Defnition 7.17 The dimension of a vector space v is the size of any basis of V. Proof of Lemma 7.15. We prove each point separately 1. If S is not linearly independent, then there are some ai such that r X #» vi = 0. ai #» i=1 Suppose WLOG that an ̸= 0. Then #» #» #» #» #» v r = a−1 r (−a1 v 1 − · · · − ar−1 v r−1 ) ∈ Span( v 1 , · · · , v r−1 ). If we take S ′ = { #v»1 , · · · , #» v r−1 }, we have Span(S ′ ) = Span(S) = V. This is because if we have a linear combination using the vectors of S, we can use the equation above to turn it into a combination of vectors in S ′ . We can repeatedly remove until we have a basis of V. Lecture 7: Fields and Vector Spaces 2. If S ⊂ Span(L), then Span(L) = V so we are done. Otherwise, suppose #» v i ̸∈ Span(L). We can create #» , .... , w #» , #» ′ v }. Then L is still linearly independent. We can just keep adding vectors to L so L′ = { w 1 s i that it stays linearly independent but eventually spans V. Pr #» is a linear combination of #v» , · · · , #» 3. Each w v . Then w ⃗ = a #v» . Let A be the r×s matrix j 1 r j i=1 ij i   a11 a12 · · · a1s  . . . . A =  ............................ . . .  ar1 ar2 ··· ars #» , · · · , w Then (w ⃗ s ) = ( #v»1 , · · · , #» v r )A.PSuppose r < s. Then by row-reduction, there exists some nonzero 1 #» #» » # vector x such that A #x» = 0 . Then xi w ⃗ i = ( #v»1 , · · · , #» v r )A #x» = 0 . This is a contradiction, since L is linearly independent, so r ≥ s. Defnition 7.18 A linear transformation is a map T :V →W such that and T ( #v»1 + #v»2 ) = T ( #v»1 ) + T ( #v»2 ) T (a #» v ) = aT ( #v»). We say that T is an isomorphism if it is a bijection (T −1 is also an isomorphism). For a vector space V over a feld F and a set of vectors S = { #v»i ∈ V }, we can defne the following linear transformation: TS : F n → V X (a1 , · · · , an ) 7→ ai #» v i ∈ V. If S is linearly independent, then TS is injective; if Span(S) = V, then TS is surjective, and if TS is a basis, then TS is an isomorphism.