Linear Algebra's Basic Concepts

We use the notation = 1, 2,..., for the positive integers. Let 0 = 0 1 2 denote the nonnegative integers, and let 0 ±1 ±2 de- , , ,..., { } = , , ,... note the set of all integers. Let × n ( -fold Cartesian product of ) be the set { } S n { S } of n-tuples from a nonempty set S. We also write Sn for this product.

Semigroup→Monoid→Group: a set with one binary operation

A function w : S2 → S is called a binary operation. It is sometimes useful to write w(x, y) in a simpler form such as x w y or simply x · y or even just x y. To tie the binary operation w to S explicitly, we write (S, w) or (S, ·). DeVnition >.> (Semigroup). Let (S, ·) be a nonempty set S with a binary operation “·” . If (x · y) · z = x · (y · z), for all x, y, z ∈ S, then the binary operation “·” is called associative and (S, ·) is called a semigroup. If two elements, s, t ∈ S satisfy s · t = t · s then we say s and t commute. If for all x, y ∈ S we have x · y = y · x then (S, ·) is a commutative (or abelian) semigroup.

Remark >.? (Semigroup). Let S = M2,2(e) be the set of 2 × 2 matrices with entries in e = 0, ±2, ±4,... , the set of even integers. DeVne w(X,Y ) = XY to be the standard multiplication of matrices (which is associative). Then { } (S, w) is a semigroup. This semigroup is not commutative. The semigroup of even integers, (e, ·), where “·” denotes multiplication of integers, is commutative.

@ Associative + Identity = Monoid

? Monoid

DeVnition ?.> (Monoid). Let (S, ·) be a semigroup. If there exists an element e ∈ S such that for all x ∈ S, e · x = x · e = x, then e is called an identity for the semigroup. A semigroup with an identity is called a monoid. If x ∈ S and there is a y ∈ S such that x · y = y · x = e then y is called an inverse of x.

Remark ?.? (Monoid). The identity is unique (i.e., if e and e0 are both identities then e = e · e0 = e0). Likewise, if y and y0 are inverses of x, then y0 = y0 · e = y0 · (x · y) = (y0 · x) · y = e · y = y so the inverse of x is unique. The 2 × 2 matrices, M2,2(), with matrix multiplication form a monoid (identity I2, the 2 × 2 identity matrix).

Associative + Identity + Inverses = Group

@ Group

DeVnition @.> (Group). Let (S, ·) be a monoid with identity e and let x ∈ S. If there is a y ∈ S such that x · y = y · x = e then y is called an inverse of x (see ?.>). A monoid in which every element has an inverse is a group.

Remark @.? (Group). The mathematical study of groups is a vast subject. Commutative groups, x · y = y · x for all x and y, play an important role. They are also called abelian groups. Note that the inverse of an element x in a group is unique: if y and y0 are inverses of x, then y0 = y0 · e = y0 · (x · y) = (y0 · x) · y = e · y = y (see ?.?).

Ring: one set with two intertwined binary operations

A Ring and Field

DeVnition A.> (Ring and Field). A ring, (S, +, ·), is a set with two binary operations such that (S, +) is an abelian group (“ + " is called “addition”) and (S − 0 , ·) is a semigroup (“ · " is called “multiplication”). The two operations are related by distributive rules which state that for all , , in : { } x y z S (left) z · (x + y) = z · x + z · y and (x + y) · z = x · z + y · z (right).

A Remark A.? (Notation and special rings). The identity of the abelian group (S, +) is denoted by “0” and is called the zero of the ring (S, +, ·). If (S − 0 , ·) is a monoid then + · is a ring with identity. If − 0 · is commutative (S, , ) (S , ) { } then + · is a commutative ring. If − 0 · is a group then the ring is (S, , ) (S , ) { } called a skew-Veld or division ring. If this group is abelian then the ring is { } called a Veld. Remark A.@ (Basic ring identities). We have taken the point of view that a semigroup, (S, ·), has S nonempty (>.>). Thus, the semigroup (S − 0 , ·) of a ring (A.>) has − 0 nonempty. If are in a ring + · then the S r, s,t (S, , ) { } following basic identities (in braces, plus hints for proof) hold: { } (1) r · 0 = 0 · r = 0 : If x + x = x then x = 0. Take x = r · 0 and x = 0 · r.

(2) {(−r) · s = r · (−}s) = −(r · s) : r · s + (−r) · s = 0 =⇒ (−r) · s = −(r · s).

(3) {(−r) · (−s) = r · s : Replace} r by −r in (?). Note that −(−r) = r.

Using{ the identities of} Remark A.@, you can show that if (S − 0 , ·) has an identity , then − · − for any ∈ and, taking − , − · − . e ( e) a = a a S a = e ( {e}) ( e) = e It is convenient to deVne r − s = r + (−s). Then we have t · (r − s) = t · r − t · s and (r − s) · t = r · t − s · t.

A Veld is a ring (S, +, ·) where (S − 0 , ·) is an abelian group { } Remark A.A (Ring and Field). The 2 × 2 matrices over the even integers, M2,2(e), with the usual multiplication and addition of matrices, is a noncommutative ring without an identity. The matrices, M2,2(), over all integers, is a noncommutative ring with identity. The ring of 2 × 2 matrices of ! x y the form where x and y are complex numbers is a skew-Veld but −y x not a Veld. This skew-Veld is equivalent to (i.e, a “matrix representation of”) the skew Veld of quaternions (see Wikipedia article on quaternions). The most important Velds for us will be the Velds of real and complex numbers.

B Ideal

DeVnition B.> (Ideal). Let (R, +, ·) be a ring and let A ⊆ R be a nonempty subset of R. If (A, +, ·) is a ring, then it is called a subring of (R, +, ·).A subring (A, +, ·) is a left ideal if for every x ∈ R and y ∈ A, xy ∈ A.A right ideal is similarly deVned. If (A, +, ·) is both a left and right ideal then it is a two-sided ideal or, simply, an ideal. Note that if (R, +, ·) is commutative then all ideals are two sided.

B ! Remark B.? (Ideal). The set of all matrices of the form x y is a sub- a = 0 0 ring of M2,2(). This subring, which has no identity element, is a right ideal but not a left ideal. It also has zero divisors - elements a , 0 and b , 0 such that a · b = 0. Another example of an ideal is the set of even integers, e, which is a subring of the ring of integers, (which, it is worth noting, has no zero divisors). The subring, e, is an ideal (two-sided) in . Given any integer n , 0, the set k · n | k ∈ of multiples of n, is an ideal of the ring which we denote by . Such an ideal (i.e., generated by a single ({n) = n = }n element, n) in is called a principal ideal. It is easy to see that all ideals in are principal ideals. Another nice property of integers is that they uniquely factor into primes (up to order and sign).

C Integral domain

Algebraists have deVned a number of important abstractions of the ring of integers, . We next discuss four such abstractions: integral domain, principal ideal domain (PID), unique factorization domain (UFD), and Euclidean domains - each more restrictive than the other.

Euclidean Domain =⇒ PID =⇒ UFD DeVnition C.> (Integral domain). An integral domain is a commutative ring with identity, (R, +, ·), with no zero divisors (B.?). We denote the identity of (R − 0 , ·) by 1R or, simply, 1. DeVnition{ } C.? (Characteristic of a ring). Let R be a ring. Given a ∈ R and an integer n > 0, deVne na ≡ a + a + ··· a where there are n terms in the sum. If there is an integer n > 0 such that na = 0 for all a ∈ R then the characteristic of R is the least such n. If no such n exists, then R has characteristic zero. Remark C.@ (Divisors, associates and primes). If b, c and a are elements of an integral domain R such that a , 0 and b = a · c then we say that a divides b (written a | b) or a is a divisor of b. An element u in R − 0 is an invertible element or a unit of if has an inverse in − 0 · . Two elements, R u (R , ) { } a and , of are associates in if · where is a unit. An element in b R R a = b u u { } p R − 0 is irreducible if p = a · b implies that either a or b is invertible and prime if | · implies | or | . For unique factorization domains (C.A), { } p a b p a p b p is irreducible if and only if it is prime. In the ring , the only invertible elements are +1, −1 . The only associates of an integer n , 0 are +n and − . The integer 12 3 · 4 is the product of two non-invertible elements so n { =} 12 is not irreducible (i.e., reducible) or, equivalently in this case, not a prime. The integer 13 is a prime with the two associates +13 and −13. A Veld is an integral domain in which every nonzero element is invertible. In a Veld,

C ˙ if 0 , p = ab then both a and b are nonzero and hence both are invertible (and "at least one is invertible" is satisVed) which implies that every nonzero element in a Veld is irreducible.

DeVnition C.A (Unique factorization domain). An integral domain R is a unique factorization domain (UFD) if

(>) Every nonzero and non-invertible a ∈ R can be factored into a Vnite product of irreducibles (C.@).

(?) If a = p1 ··· pr and a = q1 ··· qs are two such factorizations then r = s and the qi can be reindexed so that pi and qi are associates for i = 1,..., s. Remark C.B (Unique factorization domains). The integers, , are a unique factorization domain. Every Veld is also a unique factorization domain (r = s = 1 in (?) of C.A). If R is a UFD then so are the polynomial rings R[x] and R[x1,..., xn]. If a1,..., an are nonzero elements of a UFD, then there exists a greatest common divisor d = gcd(a1,..., an ) which is unique up to multiplication by units.

DeVnition C.C (Principal ideal domain). An integral domain R is a principal ideal domain (PID) if every ideal in R is a principal ideal (B.?). Remark C.D (Principal ideal domains). We noted in Remark B.? that every ideal in is a principal ideal. If (F, +, ·) is a Veld, then any subring, (A, +, ·), contains a nonzero and hence invertible element a. The ideal (a) = F. There is only one ideal in a Veld and that is a principal ideal that equals F. Thus, any Veld F is a PID. Let a1,..., an be nonzero elements of a PID, R. It can be shown that if d = gcd(a1,..., an ) in R then there exists r1,...,rn in R such that r1a1 + ··· +rn an = d. The ring of polynomials in two variables (or more) over a Veld, F[x1,... xn], is not a PID. Also, the ring of polynomials with integral coeXcients, Z[x], is not a PID.

D Euclidean domain

DeVnition D.> (Euclidean valuation). A function ν from the nonzero elements of an integral domain R to the nonnegative integers, N0, is a valuation on R if

(>) For all a, b ∈ R with b , 0, there exist q and r in R such that a = b · q + r where either r = 0 or ν(r) < ν(b).

(?) For all a, b ∈ R with a , 0 and b , 0, ν(a) ≤ ν(a · b). DeVnition D.? (Euclidean domain). An integral domain R is a Euclidean domain if there exists a Euclidean valuation on R (see D.>)

D Remark D.@ (Euclidean domains). For the three integral domain types just discussed, it can be shown that every Euclidian domain is a principal ideal domain and every principal ideal domain is a unique factorization domain. The integers are a Euclidean domain with ν(n) = |n|. The polynomials with real numbers as coeXcients, [x], form a Euclidean domain with ν(p(x)) the degree of p(x). Any Veld (F, +, ·) is a Euclidean domain with ν(x) = 1 for all nonzero x. But, the ring of polynomials with integral coeXcients, [x], is not a PID (C.D) and thus not a Euclidean domain. Likewise, the ring of polynomials in n variables, n > 1, over a Veld F, F[x1,..., xn], is not a PID(C.D) and hence not a Euclidean domain. Rings that are PIDs but not Euclidean domains are rarely discussed (the ring [α] = a+bα | a, b ∈ , α = 1 + 19 1/2 is an example). ( ( ) i) { } We now combine a ring with an abelian group to get a module.

E Module

DeVnition E.> (Module). Let (R, +, ·) be a ring with identity 1R. Let (M, ⊕) be an abelian group. We deVne an operation with domain R × M and range M which for each r ∈ R and x ∈ M takes (r, x) to r x (juxtaposition of r and x). This operation, called scalar multiplication, deVnes a left R-module M if the following hold for every r, s ∈ R and x, y ∈ M:

(1) r(x ⊕ y) = r x ⊕ ry (2)(r + s)x = r x ⊕ sx (3)(r · s)x = r(sx)(4) 1R x = x. Usually, we simply say “M is an R-module,” the “left” being understood. We also use “+” for the addition in both abelian groups and replace “·” with juxtaposition. Thus, we have: (2)(r + s)x = r x + sx (3)(rs)x = r(sx). What we call a “module” is sometimes called a “unitary module.” In that case, a “module" does not need to have an identity, 1R. Remark E.? (Module). Let R be the ring of 2 × 2 matrices over the integers, M2,2(). Let M be the abelian group, M2,1(), of 2 × 1 matrices under addition. Then (>) and (?) correspond to the distributive law for matrix multiplication, (@) is the associative law, and (A) is multiplication on the left by the

2 × 2 identity matrix. Thus, M2,1() is an M2,2()-module.

F Vector space and algebra

DeVnition F.> (Vector space and algebra). If an abelian group (M, +) is an F- module where F is a Veld (A.A), then we say (M, +) (or, simply, M) is a vector space over F (or M is an F vector space). Suppose (M, +, ·) is a ring where (M, +) is a vector space over F and where the following “scalar rule” holds,

E scalar rule: for all α ∈ F, a, b ∈ M we have α(a · b) = (αa) · b = a · (αb).

Then (M, +, ·) is an algebra over F (or M is an F algebra).

Remark F.? (Complex matrix algebra). Let ¼ be the Veld of complex numbers and let M be M2,2(¼), the additive abelian group of 2 × 2 matrices with complex entries. Conditions (>) to (A) of E.> are familiar properties of multiplying matrices by scalars (complex numbers). Thus, M is a complex vector space or, alternatively, M is a vector space over the Veld of complex numbers, ¼. If we regard M as the ring, M2,2(¼), of 2 × 2 complex matrices using the standard multiplication of matrices, then it follows from the deVnitions of matrix multiplication and multiplication by scalars that the scalar rule of F.> holds, and M2,2(¼) is an algebra over ¼.

>= Notational conventions

Remark >=.> (Special notation). Let ∈ , ¿[x] where denotes the integers and ¿[ ] the polynomials over a Veld ¿ which will be either (rational x { } numbers), (real numbers) or ¼ (complex numbers). Thus, ¿ ∈ , , ¼ . Note that is a Euclidean domain. A general theorem in algebra says that { } any integral domain can, like the integers, be extended to a quotient Veld. For , the quotient Veld is the rational numbers, . For the Euclidean domains ¿[x], the quotient Veld is all rational functions over ¿ (ratios of polynomials with coeXcients in ¿) which we denote by ¿(x) (parentheses instead of ¿[x]).