Notes on Algebra [version 30 Jun 2013 ed.]

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NOTES ON ALGEBRA Stephan Foldes Tampere University of Technology 2013 These notes are made available to students and faculty of the University of Miskolc for their personal use and as instructional material to be used at the University of Miskolc. Contents 1 General algebraic constructs 2 Linear algebra over arbitrary fields 3 Finite fields TOPIC 1: GENERAL ALGEBRAIC CONSTRUCTS SETS AND FUNCTIONS Basic familiarity with sets and functions is presupposed. A function f , or synonymously map, assigns to every element x of a specified set D, called its domain, some object f (x), which is called the value of the function at x. The set {f (x) : x ∈ D} is called the range, or image, of the function, denoted Im f . We say that the function f maps the set D to the set C, or that f is a function from D to C, if D is the function’s domain and Im f ⊆ C. We denote this fact by f : D −→ C. For x ∈ D and y ∈ C, as an alternative notation to f (x) = y, we can also write f : x 7→ y, or simply x 7→ y if f is understood from the context. The set of all functions from D to C is denoted by C D . By a famil y of objects (ai )i∈I = (ai : i ∈ I) indexed by a set I we mean nothing else but a map whose domain is the index set I and where for each i ∈ I, i 7→ ai . If the index set is a Cartesian product J × K of sets, the double index notation (aij )j∈J,k∈K may also be used. More generally, if the index set is a Cartesian product of several sets, we can use multiple indexing. The composition of two maps is denoted by g ◦ f or simply gf , where f is the map first applied. Sometimes the function value f (x) is denoted simply 1

by f x. Other notations such as f x , fx or xf are more unusual but may be convenient in certain contexts. MONOIDS, RINGS AND FIELDS Basic familiarity is assumed with semigroups, monoids, groups, and elementary ring theory, including ideals in commutative rings and polynomials. In a monoid with neutral element e, if for some element a, b we have ab = e then a is called a left inverse of b and b is called a right inverse of a. A monoid element x may have neither a left inverse nor a right inverse, it may have several left inverses or several right inverses, but if it has a left inverse and also a right inverse then it only has one left inverse x′ and one right inverse x′′ and necessarily x′ = x′′ . This element x′ = x′′ is then simply called the inverse of x, and x is said to be invertible. The product of invertible elements is invertible, and the inverse of any invertible element is obviously also invertible. Submonoids of a monoid are understood to contain the neutral element of the environment monoid. Thus a monoid M may have subsemigroups that are monoids but not submonoids of M. The set of invertible elements of M is always a submonoid which is in fact a group. Rings may be non-commutative and may or may not have an identity element. In a ring R with identity 1R (usually denoted simply by 1), i.e. when the multiplicative semigroup is a monoid, the set of invertible elements constitutes the multiplicative group R∗ of units. The null element 0R = 0 is never a unit except in a trivial one-element ring where 1 = 0. A field is a non-trivial commutative ring with identity in which every non-null element a is a unit, with inverse denoted by a−1 or 1/a. Any product of the form ba−1 is also denoted by b/a. Monoid homomorphisms map the neutral element of the domain monoid to the neutral element of the codomain monoid, thus there may be semigroup homomorphisms between monoids that are not monoid homomorphisms. Ring homomorphisms do not have to map the identity element of the domain ring (if it has an identity element) to the identity of the codomain ring (even if it has one). Homomorphisms between rings with identity elements that do map identity to identity are said to be 1-preserving. Ring isomorphisms are necessarily 1-preserving. Subrings do not have to contain the identity element of the larger ring (even if it has one). Those subrings S of 2

a field that contain the identity of the field, and contain also the inverse of every invertible element of S, are called subfields. For any field F there is one and only one 1-preserving ring homomorphism h from the ring Z of integers (with usual sum and product operations) to F . The characteristic of the field is defined as the (unique) non-negative integer k such that ker h = kZ (the ideal consisting of the multiples of k). If the characteristic is not 0, then it is a prime number p and the range of the homomorphism h is a p-element subfield contained in all other subfields of F . Finite fields necessarily have non-zero, prime characteristic. In a monoid, when we speak about a product of some number n of monoid elements, what is meant is a product of the form a1 ...an where the factors ai are not necessarily distinct. The possibilities of n being 1 or 0 are also allowed. For n = 0 the product is the neutral element of the monoid. Examples of fields (0) The power set of any set S is a ring under the operation of symmetric difference as sum, and intersection as product. This ring is a field if and only if S is a singleton, in which case we have a 2-element field. All 2-element fields are isomorphic to each other. (1) If F2 is a 2-element field, then the set F22 of couples of field elements is a 4-element field F4 under addition and multiplication defined as follows: (a, b) + (c, d) = (a + c, b + d) (a, b)(c, d) = (ac + bd, ad + bc + bd) The couples (0, 0) and (0, 1) constitute a subfield isomorphic to F2 . The field F4 has two automorphisms. (2) The set C of complex numbers under the usual addition and multiplication laws forms a field, of which the real numbers R and the rational numbers Q are subfields. MATRICES Basic familiarity with matrix operations is presupposed. An n×m matrix over a ring R , where n and m are positive integers, is a map A : {1, ..., n} × {1, ..., m} −→ R In other words, it is a double-indexed family (A(i, j) : 1 ≤ i ≤ n, 1 ≤ j ≤ m) = (aij )1≤i≤n,1≤j≤m 3

of elements of R, often denoted simply by (aij ) when the range of indices is clear from the context. The ordered couple (n, m) is referred to as the size of the matrix and it is denoted n × m (read ”n-by-m”). If n = m then the matrix is said to be a square matrix, of order n. Addition of matrices (defined by adding corresponding entries) makes the set Mn,m of matrices of the same given size n × m over R into a commutative group, of which the null matrix 0nm = 0 is the zero element. Matrix multiplication AB, defined when A = (aij ) is n × m and B = (bjk ) is m × p, is given by the usual convolution formula (AB)(i, k) =

m P

A(i, j)B(j, k) for 1 ≤ i ≤ n, 1 ≤ k ≤ p

j=1

(Readers familiar with categorical terminology may note that the set of all matrices over R is then the arrow set of a small category with the square indentity matrices as identity arrows.) The transpose of an n × m matrix A is the m × n matrix A⊤ given by A⊤ (i, j) = A(j, i)

1 ≤ i ≤ m, 1 ≤ j ≤ n

(Transposition is an involutive and identity-preserving functorial isomorphism between the category of all matrices over R and its dual category.) For any fixed n ≥ 1, matrices of size 1 × n are called row vectors, their transposes are called column vectors. Both are in obvious natural bijection with the set Rn of n-tuples of ring elements. A matrix is symmetric if it coincides with its transpose. The set of 1 × 1 matrices (all of which are symmetric) is in obvious natural bijection with R. Of particular interest are matrices over fields. Square matrices of a given size n × n over a given ground ring or field R form a semigroup under matrix multiplication, and as matrix product distributes over matrix adition, they form a ring denoted Mn (R) or simply Mn if the base ring is understood. This matrix ring has an identity if and only if R does, in that case the identity matrix is denoted by In or sinply I. For an n × m matrix, where n and m are positive integers, we also make use of the customary display of entries in tabular form a11 . . an1

. . . .

. . . .

. . . . 4

. a1m . . . . . anm

Example Over a 2-element field, the ring of 2 × 2 matrices has 16 elements. The following four matrices constitue a subring which is in fact a field:         0 1 1 1 1 0 0 0 1 1 1 0 0 1 0 0 MODULES, VECTOR SPACES AND SUBSPACES Let R be a commutative ring with an identity element denoted 1. A module over R is specified by a commutative group V , with operation denoted additively, and a map R × V −→ V associating to each ring element a ∈ R and v ∈ V an element av of V , such that for all a, b ∈ R and v, w ∈ V , a(bv) = (ab)v (a + b)v = av + bv a(v + w) = av + aw 1v = v With 0 denoting the null element of the additive group V , it follows that 0v = 0 = a0 (−a)v = −(av) = a(−v) (−1)v = −v A module over a field F is called a vector space. The elements of F are called scalars, those of V are called vectors, and for every scalar a and vector v their product is the vector av. This product is null only if both a and v are null. When F and the map (a, v) 7→av are understood, we refer to V as ”the vector space V ”. The vector space is called trivial when it is reduced to the null vector. Particular vector spaces are usually specified by providing the rules for addition of vectors and for computing av from a and v. Examples (1) For any field F and integer n ≥ 0 the set F n of all ntuples (a1 , ..., an ) of elements of F is a vector space over F where the vector addition group is defined by (a1 , ..., an ) + (b1 , ..., bn ) = (a1 + b1 , ..., an + bn ) 5

and for any a ∈ F

a(a1 , ..., an ) = (aa1 , ..., aan )

(2) For any set S, the power set P(S) is a vector space over a 2-element field F2 (any 2-element field) with vector addition defined as symmetric difference. This specifies the vector space structure completely in this case, because there can be only one 1-preserving ring-homomorphism from F2 . Within P(S) the set of finite subsets of even cardinality also constitutes a vector space over F2 , again with symmetric difference as vector addition. (3) The set of matrices over a field F having a given size m×n is a vector space with usual matrix sum as vector addition, and where for any scalar c ∈ F and matrix A = (aij ) the product cA is given by the multiplication of each entry by the scalar, cA = (caij ) (4) Let F be any field, and N any set. The set F N of all functions from N to F is a vector space with sum and scalar-vector product given by (f + g)(x) = f (x) + g(x) (af )(x) = a(f (x)) In some sense this example encompasses the previous examples. In what sense? (5) The set C of complex numbers is a vector space over the real field R: additon of vectors is complex addition and the product of a scalar with a vector is the complex product of the scalar as a real and the vector as a complex number. In general, if R is any ring with identity and F is any subring that is a field, then R is a vector space over F with addition of vectors coinciding with the sum operation in R, and multiplication of a scalar with a vector coinciding with their product in R. In some sense this example subsumes all the previous examples. In what sense? A (linear ) subspace of a vector space V over a field F is a subgroup U of the additive group V such that av ∈ U for all a ∈ F, v ∈ U. Equivalently, it is a non-empty subset U of V such that for all a, b ∈ F and v, w ∈ U, the vector av + bw

6

also belongs to U. In this case U has a natural vector space structure over F inherited from V in which the product of a scalar a and a vector v ∈ U is their product av as defined in the environment space V. Examples and counter-example (1) In a finite power set vector space over a 2-element field, the set of even subsets is a subspace. (2) In the space of complex numbers over the real field, the set of Gaussian integers (i.e. complex numbers with integer real and imaginary parts) is not a subspace. It is a module, however, over the ring Z of integers, the product of an integer a ∈ Z and a Gaussian integer z = c + di being the Gaussian integer ac + adi. (3) The set of symmetric square matrices of a given size over a field F is a subspace of the space of all square matrices of the same size over F . RINGS AND NON-ASSOCIATIVE ALGEBRAS It sometimes makes sense to consider ring-like structures in which the multiplication is not necessarily commutative or associative. We shall examine the following concept of algebra, which is not the most general possible but general enough to encompass some very different classical algebraic structures. We shall call algebra a commutative group A whose operation is called addition or sum and is denoted additively, together with a binary operation called multiplication or product, denoted by juxtaposition or ·, and such that for all a, b, c ∈ A we have a(b + c) = ab + ac (b + c)a = ba + ca In general, the product operation does not have to be commutative or associative. Examples (1) Any ring. (2) For any positive integer n, the set of n × n real matrices with matrix sum and ordinary matrix product. (3) The set of n × n real matrices with matrix sum and the product of matrices M, N defined as their commutator MN − NM. Subalgebras are defined as subgroups of an algebra closed under multiplication, they constitue a closure system on any given algebra A. 7

A set C ⊆ A is said to have the coefficient property with respect to S ⊆ A if for all c, d ∈ C and s, t ∈ S we have (cs)(dt) = (cd)(st). Algebra homomorphisms are defined as additive group homomorphisms preserving also multiplication. The inverse of any bijective homomorphism is clearly also a homomorphism, called isomorphism of algebras. A set S together with a partially defined binary operation ω : D −→ S, D ⊆ S 2 , will be called a partial groupoid. If (a, b) ∈ D then for the value ω(a, b) we generally use the infix notation aωb, or simply the juxtaposition ab. and by S we may denote not just the underlying set but also the partial groupoid when ω is understood from the context. An element e ∈ S is called neutral (or an identity) if whenever ea or ae is defined, it is equal to a. Note that if the product of two neutral elements is defined, then the two elements are equal and they are also equal to their product. Every subset S of an algebra A has a natural partial groupoid structure whose domain of operation is that of the product operation of A restricted to  (a, b) ∈ S 2 : ab ∈ S

By a homomorphism from a partial groupoid S to a partial groupoid T we mean a map f : S −→ T such that whenever xy is defined in S the product f (x)f (y) is defined in T and equals f (xy). The inverse map of a bijective homomorphism is then not necessarily a homomorphism, but if it happens to be then we speak of isomorphisms and isomorphic.partial groupoids. A partial multiplicative graph, or non-associative partial category, is a partial groupoid S such that for every element a ∈ S there are unique identities e and e′ such that ea and e′ a are defined (and they are then equal to a). We say that these identities are the source and the target of the arrow a, respectively, and we denote them by source(a) and target(a), and we also require that ab be defined only if target(a) = source(b) and then source(ab) = source(a) and target(ab) = target(b). We speak of a non-associative category if ab is always defined whenever target(a) = source(b). If in addition (ab)c = a(bc) holds whenever ab an bc are defined, we speak of a category. 8

For any partial groupoid S and any algebra A, let ASfin denote the set of almost everywhere zero maps from S to A (i.e. maps that take the value 0 on the complement of some finite subset of S). This set is given an algebra structure by defining p + q and pq by (p + q)(a) = p(a) + q(a) X (pq)(a) = p(b)q(c) bc=a

for all a ∈ S. This algebra is called the partial groupoid algebra of S over A. If S is a monoid, a group, a category, then we speak of a monoid algebra, a group algebra, a category algebra. Any binary relation S ⊆ E 2 on a set E is a partial groupoid where (u, v)(z, w) is defined if and only if v = z and (u, w) ∈ S, and then (u, v)(z, w) = (u, w) The algebra of the binary relation (over a given base ring) is then called the incidence algebra of the binary relation (over that base ring). If S is reflexive then it is a non-associative partial category. If in addition S is transitive, then it is a category. Examples (1) The ring of polynomials (in one indeterminate) over a base ring A is isomorphic to the monoid algebra of the additive monoid N over A. (2) The non-commutative ring of n × n matrices over a ring A is isomorphic to the incidence algebra of the full binary relation {1, ..., n}2 on {1, ..., n} . Algebra Extension Theorem Let A be an algebra with multiplicative identity element 1 6= 0 Let S be a partial multiplicative graph with only finitely many identities. For every t ∈ A denote by t′ the map in ASfin which to every identity associates t and which maps every other element of S to 0.Then A is isomorphic to a subalgebra A′ of ASfin via the embedding t 7−→ t′ . For every s ∈ S denote by s′ the map in ASfin which to s associates 1 and otherwise maps everything to 0 and denote by S ′ the set of elements s′ so obtained..The set S ′ with multiplication inherited from A is a partial groupoid isomorphic to S via s 7→ s′ . The algebra ASfin coincides with A′ [S ′ ] and A′ has the coefficient property with respect to S ′ .Every p ∈ ASfin can be written as

9

p=

P

[p(s)]′ s′

s∈S p(s)6=0

 MATRICES AND POLYNOMIALS In this section we shall consider polynomials over any ring R that is not necessarily commutative. The definition of polynomials is the same as in the commutative case, and the polynomials form a ring for which the notation R[X] is continued to be used. For any field F and n ≥ 1, the set Mn (F ) of n×n matrices is a ring under matrix addition and product (not commutative if n > 1). Over this ring we can construct the ring Mn (F )[X] of polynomials with matrix coefficients. On the other hand, we can construct the ring Mn (F [X]) of matrices with polynomial entries. Commutation Lemma (Matrix and Polynomial Ring Constructions) For any field F and n ≥ 1, the rings Mn (F )[X] and Mn (F [X]) are isomorphic via A0 + A1 X + ... + Am X m 7→ B where the entry B(i, j) is A0 (i, j)+A1 (i, j)X +...+Am (i, j)X m .



Any field F is canonically embedded to the matrix ring Mn (F ) by a 7→ diag(a, ..., a) The range of the embedding can be identified (confused) with and denoted by F , each matrix diag(a, ..., a) being identified with and denoted by a. Each polynomial over F is thus identified with a polynomial over Mn (F ). There is a unique ring homomorphism F [X] −→ Mn (F )Mn(F ) ,

b 7→fb

such that (i) for each constant polynomial a, fa is the constant map with value a, and (ii) fX is the identity map on Mn (F ). For any matrix C ∈ Mn (F ), we denote by

10

Recall that for every integral domain D there is a field F containing D as a subring, or equivalently, for every integral domain D there is a field F and an injective ring homomorphism D → F . Such a field F can be constructed as follows. First two commutative monoid structures, one with additive and another one with multiplicative notation, are defined on the set D 2 by (a, b) + (c, d) = (ad + bc, bd) (a, b)(c, d) = (ac, bd) The set D1 ⊂ D of couples (a, b) such that b 6= 0 constitutes both an additive and a multiplicative submonoid. The equivalence relation ≡ on D1 defined by (a, b) ≡ (c, d) ⇐⇒ ad = bc is a congruence of both the additive and multiplicative monoid structures on D1 .The additive quotient monoid D1 / ≡ is a commutative group, and together with the multiplicative quotient operation on D1 / ≡ it defines the addition and product operations of a field F = D1 / ≡. The injection D → F is given by a 7→ (a, 1). A PROOF OF HILBERT’S FINITE BASIS THEOREM (TRANSFER THEOREM) By a ring we shall now mean a commutative ring with multiplicative identity denoted by (some version of) the symbol 1. Basic ideal theory is assumed to be known. A ring is said to be Noetherian if the following equivalent conditions hold: (i) every ideal I has a finite set of elements that generate I, (ii) every non-empty set S of ideals has a member that is maximal with respect to inclusion (i.e. that is not contained in any other member of S), (iii) there is no infinite ascending sequence of ideals J0 ⊂ J1 ⊂ ... ⊂ Jn ⊂ Jn+1 ⊂ ... Examples The ring Z of integers. Any field. Any finite ring, in particular the proper quotients of Z. Hilbert’s Finite Basis Theorem The ring A[X] of polynomials over a Noetherian ring A is Noetherian. 11

Proof Suppose there is in A[X] an ascending sequence of ideals {0} = J0 ⊂ J1 ⊂ ... ⊂ Jn ⊂ Jn+1 ⊂ ... and let J denote the union of this sequence of ideals - this is also an ideal. The sequence of numbers mn = min {do p : p ∈ Jn \ Jn−1 } , n = 1, 2, ... contains a non-decreasing infinite subsequence. Let σ be the supremum (possibly ∞) of the terms of this subsequence. The ideal M of A consisting of 0 and the leading coefficients of all nonnull polynomials in J of degree not exceeding σ has a finite generating set consisting of k ≥ 1 non-zero elements a1 , ..., ak . Let q1 , ..., qk be polynomials in J of degrees d1 , ..., dk not exceeding σ, with leading coefficients a1 , ..., ak , and let m be the maximum of the degrees d1 , ..., dk . There exists an n ≥ 1 such that these polynomials all belong to Jn . Let l be the first integer larger than n such that σ ≥ ml ≥ m, let q be a polynomial of degree ml in Jl \ Jl−1 and let a denote its leading coefficient. As a ∈ M, there are c1 , ..., ck such that a = c1 a1 + ... + ck ak Then q − (c1 q1 X ml −d1 + ... + ck qk X ml −dk ) ∈ Jl cannot belong to Jl \ Jl−1 , it must belong to Jl−1 , but then so does q, which is contrary to assumption.  TOPIC 2: LINEAR ALGEBRA OVER ARBITRARY FIELDS 2.1 INDEPENDENCE AND DIMENSION: A MATROID APPROACH We develop here a slight generalization of the rudiments of matroid theory, a branch of combinatorics. The generalization consists in considering not only finite, but also infinite underlying sets V , such as vector spaces over arbitrary fields. A closure system on a set V is a set of subsets of V , called the closed subsets, such that (i) V is closed, (ii) the intersection of any non-empty set of closed sets is closed.

12

The closure of any subset S of V is then the intersection of all closed sets containing S, this closure is in fact the smallest closed subset containing S, and it is denoted by S. We also say that S generates, or spans, S. The map S → S on P(V ) is a closure operator, i.e. for all S, T ⊆ V we have (i) S ⊆ S (ii) S = S (iii) S ⊆ T =⇒ S ⊆ T An element v of V is said to depend on S, if v belongs to the closure of S. The set S is independent if no v ∈ S depends on S \ {v}, otherwise S is dependent. A minimal dependent set is called a circuit. It is easy to see that every subset of an independent set is independent. And importantly, the empty set is always independent. Examples (0) If V is any vector space, the set of subspaces constitutes a closure system. (1) Let V consist of the set of all point of the plane, and let the closed subsets be specified as the empty set, the one-point singletons, all lines (a line being thought of as the set of points that it consists of), and the entire set V . Now any two distinct points generate a line, and three distinct points are independent or constitute a circuit, according to whether they form a triangle or are collinear. There are also circuits consisting of 4 points. (2) Let V be the plane as before and let the closed sets be specified as the convex sets of points. (3) Let V consist of the set of edges of a finite graph (undirected, possibly with loops and multiple edges). Every set S of edges defines a subgraph on the same set of vertices as the whole graph – consider a set S closed if no larger set of edges S ′ ⊃ S defines a subgraph with the same number of connected components. Now a set S is independent if and only if the subgraph it defines is a forest, and circuits correspond to circuits in the sense of graph theory, including single-edge loops. (4) Let V be a topological space, and let the closed sets be those that are topologically closed. Now a set S is independent if and only if each of its points has a neighborhood excluding all the other points. In the case V = R with the usual topology of the real line, an element depends on a set S if and only if it is the limit of a convergent sequence extracted from S. At the risk of some confusion with arithmetical operations, if S is any set and x an element that may or may not belong to S, we shall sometimes 13

write S + x and S − x for the sets S ∪ {x} and S \ {x} . A closure system is said to have the exchange property if the following holds: whenever x an y are elements not depending on a set S, the element x depends on S + y if and only if y depends on S + x. In the Examples above, the closure systems described in (0), (1) and (3) have the exchange property. (What about the others?) The reader can now prove the first results of linear algebra in the rarified language of matroid theory: Circuit Symmetry If the exchange property holds in a closure system, then every element of a circuit depends on the other elements of the circuit.  Property of Intersecting Circuits In any closure system with the exchange property, if C1 and C2 are two distinct circuits and v ∈ C1 ∩C2 , then (C1 ∪C2 )\{v} is a dependent set. . Minimax Characterization of Bases If the exchange property holds in a closure system on a set V , then the following conditions are equivalent for any set B of elements: (i) B is a maximal independent subset of V , (ii) B is a minimal generating set of V , (iii) B is independent and generates V .  If the above equivalent conditions hold for a subset B of V with respect to a closure system on V having the exchange property, then we call B a basis. Given any closed subset U of V , we get an inherited closure system on U whose closed sets are the closed sets of the original system contained in U. If the original system has the exchange property, then so does the inherited system. A subset of U that is a basis in the inherited system is called simply a basis of U. The independent sets, dependent sets and circuits of the inherited system are exactly those independent sets, dependent sets and circuits of the original system that are contained in U. A closure system is said to be algebraic if the closure of any set S is the same as the union of the closures of all the finite subsets of S. Equivalently, this means that any element depending on a set S depends on some finite subset of S. This has the following consequence in terms of chains of sets 14

(sets of sets any two of which are comparabble by inclusion): in an algebraic closure system, the union of any non-empty chain of closed sets is closed. (In fact this is another equivalent characterization of algebraicity.) Also, in an algebraic closure system the union of any non-empty chain of independent sets is independent. For finite chains of sets these properties are of course trivial. We shall call any algebraic closure system with the exchange property a matroid. (In the literature this term is often understood to imply the underlying set is finite, but we shall not make this restriction.) Examples The subspace closure system on any vector space constitutes a matroid. So does the closure system of points and lines of the plane (in which the empty set and the full plane are also closed.) The best counterexample to the exchange property is the convex closure system on the plane, and the classical counterexample to algebraicity is topological closure. The next two theorems about algebraic closure systems are easily proved by elementary set-theoretical reasoning: Finiteness of Circuits In any algebraic closure system, all circuits are finite.  Circuit Characterization of Independence In an algebraic closure system, a set is independent if and only if it does not contain any circuit. Therefore a set is independent if and only if all of its finite subsets are independent.  Maximal Independent Set Theorem (i) Every algebraic closure system has a maximal independent set. (ii) Every matroid (i.e. algebraic closure system with the exchange property) has a basis. Proof It is sufficient to prove (i), statement (ii) is just a re-wording for matroids. First, observe that the assertion is trivial if the underlying set of the closure system is finite. Secondly we prove the theorem for closure systems in which there is no infinte ascending chain C1 ⊂ C2 ⊂ ...of closed sets (”the Noetherian property”). In this case there can be no infinite independent set (because within 15

this an ascending chain of larger and larger independent sets would generate an ascending chain of closed sets). But then in trying to enlarge any independent set (say, the empty set) into larger and larger independent sets, we must at some point find a maximal one which can no longer be enlarged. Thirdly, we remove the restrictive Noetherian hypothesis and invoke Zorn’s Lemma, which says that if in a non-empty partially ordered set P every chain has an upper bound, then P has a maximal element. We apply this to the set of all independent sets partially ordered by set inclusion. Zorn’s Lemma guarantees the existence of a maximal independent set.  The following statement is more general than statement (ii) above, it can be proved similarly: Basis Sandwich Theorem In any matroid, if I is any independent set and G is a generating set such that I ⊆ G, then there is a basis B such that I ⊆ B ⊆ G.  Corollary In any matroid, every independent set can be expanded to a basis, and every generating set can be cut down to a basis.  The Basis Sandwich Theorem will also allow us to prove that all bases in any algebraic closure system with the exchange property have the same cardinality, and on this result the notion of dimension will be based. As an intermediate step we establish the following: Steinitz Exchange Lemma In any matroid, let I, J be independent sets none of which contains the other. (i) For every v ∈ I \ J there is some w ∈ J \ I such that the set J+v − w is independent. (ii) For every w ∈ J \I there is some v ∈ I \J such that the set J+v − w is independent. Proof We prove only (i), statement (ii) can be proved similarly. Unless J + v is independent (in which case we are finished), there is a circuit C in J + v containing v. There is only one such circuit because if K were another one, then (C ∪K) −v ⊆ J would be dependent. Clearly C −v is not a subset of I, because I is independent. Take then any vector w ∈ (J \ I) ∩ C 16

The set (J − w) + v cannot contain any circuit K, for such a K would necessarily be a circuit distinct from C in J + v containing v, which we just ruled out.  Basis Equicardinality Theorem All the bases of any given matroid have the same cardinality. Proof Suppose that some basis A has a smaller cardinality than some other basis B. Call a map f from a subset S ⊆ A \ B to B \ A an exchange function if (B \ Im f ) ∪ S is independent. The set of all exchange functions is partially ordered by restriction, and by Zorn’s Lemma there is an exchange function f which is not a restriction of any other exchange function. (Actually Zorn’s Lemma is not needed if A is finite.) This function f cannot be surjective to B \ A because CardA < CardB. Its domain S cannot be all of A \ B because then A would be a subset of the necessarily strictly larger independent set (B \ Im f ) ∪ S, which is impossible since A is a basis. Now applying the Steinitz Exchange Lemma to I = A, J = (B \ Im f ) ∪ S and any v in I \ J = (A \ B) \ S,take any w in J \ I = (B \ A) \ Im f and extend f by v 7→ w to a larger exchange function, contradicting the maximality of f.  In a matroid the cardinality of a basis is called the matroid’s rank or dimension. For any closed subset U of the underlying set V , the dimension of the inherited closure system on U is called the rank or dimension of U, denoted dim U. Further, the rank of any subset S ⊆ V , closed or not, is defined as the dimension of its closure S and it is denoted by rank S. Equivalently, the rank of a set S is the cardinality of any maximal independent set contained in S. Obviously the rank of a set cannot exceed its cardinality. Characterization of Independence and Closure by Rank In any matroid (i) a finite set of elements is independent if and only if its rank equals the number of its elements, (ii) a set S of finite rank is closed if and only if enlarging it by adding any new element to it always increases its rank, (iii) for any closed set U of finite rank and any subset S ⊆ U, the set S generates U if and only if its rank equals that of U.  17

The Basis Sandwich Theorem yields: Rank Monotonicity In any matroid, if U and W are closed sets such that U ⊂ W , then dim U ≤ dim W. If dim U is finite, then U ⊂ W actually implies dim U < dim W. For any sets of elements S ⊆ T, we have rank S ≤ rank T.  Application to vector spaces, and linear algebra terminology This generalized matroid theory applies to the closure system of all subspaces of a vector space V over any field F : as noted above, this closure system is algebraic and possesses the exchange property. A vector w is said to be a linear combination of vectors v1 , ..., vn , n ≥ 0, if there are scalars a1 , ..., an such that w = a1 v1 + ... + an vn

(2)

For n = 0 this sum is interpreted as the null vector. For any set S of vectors, the subspace generated by S consists of all possible linear combinations of vectors taken from S. A set of vectors generating a subspace is also said to span that subspace. A vector depends on a set S of vectors, if and only if it can be written as a linear combination of some members of S: in this case we also say that the vector linearly depends on S. An independent set of vectors is also said to be linearly independent. A set I of vectors is linearly independent if and only if for all n ≥ 1 and distinct vectors v1 , ..., vn ∈ I, every vector w can be written as a linear combination (2) in at most one way. Equivalently, a set I of vectors is linearly independent if and only if for all n ≥ 1 and distinct vectors v1 , ..., vn ∈ I, the null vector w = 0 can be written as a linear combination (2) only in the trivial way with a1 , ..., an all equal to 0. Finally, distinct non-null vectors v1 , ..., vn , n ≥ 0, belonging a subspace U, constitute a basis of U if and only if every vector w ∈U can be written as a linear combination (2) in one and only one way. The null vector cannot belong to any basis, and it is in fact the only vector that by itself constitutes a circuit. Two distinct non-null vectors v1 and v2 constitute a circuit if and only if there is a scalar a such that v1 = av2 , this scalar is then necessarily non-zero and unique, and we say that the vectors v1 and v2 are proportional. By bases and dimension of a vector space we mean the bases and dimension of the closure system of subspaces. For any field F and integer n ≥ 0 the vector space F n has dimension n: the n vectors having a single 18

component equal to 1 and all other components euqal to 0, called standard unit vectors, constitute a basis of F n .This basis is referred to as the standard basis. The standard unit vectors are customarily denoted by e1 , ..., en , where ei = (δi1 , ...δin ), using the Kornecker delta notation. According to whether the dimension of a vector space V is finite or infinite, V is called finite dimensional or infinite dimensional, respectively. If we wish to emphasize the field F of scalars, we speak of dimension over F and write dimF V for dim V. This can be helpful if F has a proper subfield and we wish to consider V also as vector space over that subfield. For example, while for every n ≥ 0 the real vector space Rn is n-dimensional, dimQ Rn is infinite. Much of classical linear algebra is finite dimensional, and so is most of the linear algebra of vector spaces over finite fields. The dimension of a subspace U of a finite dimensional vector space V is at most the dimension of V , with equality if and only if U = V. Application to affine geometry On any vector space V there is a closure system larger than that consisting of the subspaces, namely the set of all additive cosets of subspaces (sets of vectors the form v + U = {v + u : u ∈ U} where U is any subspace) plus the empty set. Altogether these are called affine flats or affine subspaces of V , constituting the affine closure system on A set A of vectors is a flat if and only if for all u, v, w ∈ A and scalars a, b, c with a + b + c = 1 the vector au + bv + cw is also in A, or equivalently, if and only if for all n ≥ 1 the set A contains all possible linear combinations a1 v1 + ... + an vn of members of A with a1 + ... + an = 1. Linear combinations with coefficients summing to 1 are called affine combinations. The affine flat generated (spanned) by any set S of vectors consists of all affine combinations of members of S. The affine closure system is also algebraic and possesses the exchange property, and thus constitutes a matroid. In the affine context, it is customary to speak of points instead of vectors, and to refer to the closure of a set as affine hull, and to dimension of a flat as affine dimension. In this closure system, a point depends on a set S of points, if and only if it is an affine combination of some members of S. If a set of points is independent, these points are customarily said to be in general position. The empty set and all singletons are closed, and also independent, and these are the only sets of affine dimension 0 and 1. MINKOWSKI SUMS, JOINS AND COMPLEMENTS 19

Given any closure system on a set V , the join of two closed sets C and K is the closed set C ∪ K, ususally also denoted by C ∨ K. We say that K is a complement of C if C ∨ K = V (the largest closed set) and C ∩ K = ∅ (the smallest closed set, called also the trivial closed set). Obviously then C is also a complement of K, so we also say that C and K are complementary. A closed set may well have no complement, or several complements. Note that V and the trivial closed set are unique complements of each other. An element x ∈V is called a loop if the singleton {x} is a circuit. It is easy to see that the set of all loops is precisely ∅, the smallest closed set. Two distinct elements x, y are said to be parallel if {x, y} is a circuit. Examples In the closure system of all subspaces of a vector space, the trivial closed set is the null subspace consisting of the null vector alone, and two vectors are parallel if they are non-null but one is a scalar multiple of the other. In the closure system of all affine flats in a vector space, the trivial closed set is the empty set. Trivial Intersection Property In any algebraic closure system with the exchange property, let I and J be two disjoint independent sets. Then I ∪ J is independent if and only if I ∩ J = ∅. Proof The ”if” part is easily verified. To prove the ”only if” part, suppose that I ∪ J is independent. If I ∩ J contained an element v that is not a loop, then there would be circuits C ⊆ I ∪ {v} and K ⊆ J ∪ {v}. Since both C and K would have to contain v, by the Property of Intersecting Circuits (C ∪ K) \ {v} = I ∪ J would be dependent, which is contrary to assumption.  The proofs of the following two theorems rely on the Basis Sandwich Theorem, we leave the first one to the reader: Existence of Complementary Flats In any matroid, every closed set has at least one complement .  Dimension Submodularity In any matroid the join U ∨ W of any two finite dimensional flats U and W is also finite dimensional and dim(U ∩ W ) + dim(U ∨ W ) ≤ dim U + dim W 20

Proof Take any basis B of U ∩W. Using the Sandwich Theorem, expand B to a basis B1 of U. Separately, expand B to a basis B2 of W . As B1 ∪ B2 generates U ∨W , it must contain a basis B3 of U ∨W . Observe that B1 ∩B2 = B and count.  Rank Submodularity In any matroid, if two sets S, T have finite rank, then their union S ∪ T is also of finite rank and we have rank (S ∩ T ) + rank (S ∪ T ) ≤ rank S + rank T Proof Similar to the proof of Dimension Submodularity, and using the fact that S ∩ T ⊆ S ∩ T .  In a vector space the join of two subspaces has an algebraic expression, For any sets of vectors A and B the (Minkowski ) sum, denoted by A + B, is the set {a + b : a ∈ A, b ∈ B} For subspaces U and W , their sum is in fact the smallest subspace containing both, i.e. U ∨ W = U + W.

In general, strict inequalities can hold in the two theorems above. However, for subspaces of a vector space, dimension submodularity can be improved to an equality. To show this, we shall make use of the following: Trivial Intersection Property In the closure system of subspaces of any vector space, let I and J be two disjoint linearly independent sets of vectors in a vectors space. Then I ∪ J is independent if and only if the subspaces I and J that they generate only have the null vector in ciommon. I ∩ J = ∅. Proof The ”if” part is easily verified. To prove the ”only if” part, suppose that I ∪ J is independent. If I ∩ J contained a non-null vector v, then there would exist circuits C ⊆ I ∪ {v} and K ⊆ J ∪ {v}. Since both C and K would have to contain v, by the Property of Intersecting Circuits (C ∪ K) \ {v} ⊆ I ∪ J would be dependent, which is contrary to assumption.  Dimension Equation for Subspaces If U and W are finite dimensional subspaces of a vector space then dim(U ∩ W ) + dim(U + W ) = dim U + dim W 21

Proof In the proof of Dimension Submodularity, we claim that B1 ∪ B2 is independent. Suppose it contained a circuit P C. Then the null vector can be written as a linear combination 0 = cv v with all the coefficients cv v∈C P cv v belongs the subspace U and therefore non-null. It follows that v∈C\B1

to U ∩ W = B. By the Trivial Intersection Property applied to I = B and P J = C \ B1 the vector cv v must be null. This would force C ⊆ B1 v∈C\B1

which is impossible. 

An important particular case of the equation above is when the intersection U ∩ W is the trivial subspace reduced to the null vector. In that case any basis of U and any basis of W can be aggregated to a basis of U + W . Instead of two subspaces, consider now subspaces U1 , ..., Uk .for any k ≥ 0. Their union U1 ∪ ... ∪ Uk generates a subspace that we shall denote by U1 +...+Uk . (In fact we can arrive at this subspace by repeated application of the binary Minkowski sum operation which is associative and commutative.) Note that even if the summands Ui are pairwise disjoint, the sum of the dimensions can still be less than the dimension of the sum U1 + ... + Uk . Direct Sum Characterization Let U1 , ..., Uk be subspaces of a vector space, k ≥ 0.The following conditions are equivalent: (i) There are bases B1 , ..., Bk of the subspaces U1 , ..., Uk such that the union B1 ∪ ... ∪ Bk is a basis of U1 + ... + Uk . (ii) For every choice of bases B1 , ..., Bk of the subspaces U1 , ..., Uk the union B1 ∪ ... ∪ Bk is a basis of U1 + ... + Uk . (iii) For every choice of independent subsets I1 ⊆ U1 , ..., Ik ⊆ Uk the union I1 ∪ ... ∪ Ik is independent. (iv) For every 1 ≤ i ≤ k, the subspace Ui ∩(U1 + ... + Ui−1 + Ui+1 + ... + Uk ) is trivial. (v) Every vector can be written in at most one way as v1 + ... + vk with v1 ∈ U1 , ..., vk ∈ Uk (vi) Every vector in U1 + ... + Uk can be written in a unique way as v1 + ... + vk with v1 ∈ U1 , ..., vk ∈ Uk  If the equivalent conditions of the characterization above hold, the sum U1 +...+Uk is said to be a direct sum. In that case we have dim (U1 + ... + Uk ) = dim U1 + ... + dim Uk . 22

It was noted above that for a subfield F of a larger field E, any vector space V over a field E is also a vector space over F , but the dimensions over E and over F are likely to differ. It is clear, however, that dimE V ≤ dimF V , because any basis of V over E is still independent over F . In fact a sharper statement holds when we also consider E as a vector space over F : Dimension Multiplicativity over Nested Fields If V is a finite dimensional vector space over a field E, and E is finite dimensional when considered as avector space over a subfield F , then dimF E dimE V = dimF V Proof Let e1 , ..., em be the m distinct vectors in some basis of E over F . Let v1 , ..., vn be the n distinct vectors of some basis of V over E. Then the mn distinct vectors ei vj , 1 ≤ i ≤ m, 1 ≤ j ≤ n, constitute a basis of V over F.  2.2 SQUARE MATRICES AND DETERMINANTS DETERMINANTS From basic combinatorics we recall that a permutation of a set N is a bijective map σ : N −→ N, and that the set of all such permutations is a group under composition, called the symmetric group on N and denoted by SN . For N = n = {1, ..., n} we write Sn (For every finite set N there is a unique n, namely the number of elements of N, such that the group SN is isomorphic to Sn . ) The symmetric group SN is the group of invertible members of the monoid S S of all maps S −→ S. Recall that a tournament on a set N is a set A ⊂N 2 of ordered couples such that for every proper (non-singleton) pair {i, j} ⊆ N exactly one of the two couples (i, j) or (j, i) is in A. Observe that if A an B are any two tournaments on n = {1, ..., n} then Q σ(j) − σ(i) Q σ(j) − σ(i) = j−i j−i (i,j)∈A (i,j)∈B 23

We define the signature function sgn : Sn −→ Q to be this product of ratios of differences.taken over any tournament. In particular we have Q σ(j) − σ(i) Q σ(j) − σ(i) = j−i j−i i