1. MATRIX AND DETERMINANTS 1.1. Field and Number Matrix. - Goal: Conceptually speaking, matrix is describing a linear transformation when elements are in R, but matrix itself is more than that. In this section, you are going to learn the most basic, strict, or even abstract concepts of number matrix, to start with, we should define every term: Number and Matrix. To define what is numbers, we define the terminology: Fields. 1.1.1. Fields. Definition 1.1.1 We call a set F that equipped with two two-term-operations + and × a Field if this set satisfies the following axioms (1) Closed under addition and multiplication if a 2 F and b 2 F , then a + b 2 F and a × b 2 F (2) Commutativity of + For any two elements a; b 2 F , we have a + b = b + a (2) Associativity of + For any three elements a; b; c 2 F we have (a + b) + c = a + (b + c) (3) Existence of 0 There exists an element 0, such that for any elemnt a 2 F ,a + 0 = a. (4) Existence of opposite For any element a 2 F , There exists an element called −a, such that a + (−a) = 0 (5) Commutativity of × For any two elements a; b 2 F , we have a × b = b × a (6) Associativity of × For any three elements a; b; c 2 F we have (a × b) × c = a × (b × c) (7) Existence of 1 There exists an element 1, such that for any elemnt a 2 F , a × 1 = a. (8) Existence of reciprocal For any non-zero lement a 6= 0 2 F , There exists an element called a−1, such that a × a−1 = 1 (9) Distribute property for any a; b; c 2 F , we have a × (b + c) = a × b + a × c As you see, there is nothing strage for us. The importance of this axioms is that not only real number or complex numbers could be numbers. There are more other type of useful numbers A simple example of a field is the field of rational numbers, consisting of numbers which can be a written as fractions b , where a and b are integers, and b 6= 0. The opposite of such a fraction is a b simply − b , and the reciprocal (provided that a 0) is a . Suppose F only has two elements, namely 0 and 1, they are zero element and unit element respec- tively. And we define 1 + 1 = 0 ,0 + 1 = 1,0 + 0 = 0,0 ∗ 0 = 0,0 ∗ 1 = 0, 1 ∗ 1 = 1. Then this is a field. This is the minium size field in the universe — field of two element. It is clear that real numbers(all rational and irrational numbers) formed a field. Later on we will use complex numbers, which can be expressed as a + bi, where i is the mysterious element that saitisfying i2 = −1. We can verify that complex numbers also satisfies the axioms of field. In the book ,we denote real number field as R, and complex number field as C In this book, the element in field are always called number or scalar. 1 In the study of linear algebra, you will not only encounter calculation, but also proofs. proofs are logic analysis of why some statement is right based on axioms. To show an example, we proof the following proposition. Proposition 1.1 For any element a 2 F , a × 0 = 0 Proof. We proof as following: Assume c = a × 0 We Claim that c = c + c, because: By definition of 0, we have 0 + 0 = 0. Thus c = a × 0 = a × (0 + 0) = a × 0 + a × 0 = c + c So we verified c = c + c By the existance of −c, we add −c on both side of equation Thus, (−c) + c = (−c) + c + c So 0 = c Thus we proved c = 0 This means a × 0 = 0 1.1.2. Matrices. Definition 1.1.2 A Matrix M over a field F or, simply, a matrix M(when the field is clear) is a rectangular array of numbers aijin F , which are usually presented in the following form: 0 1 a11 a12 ··· a1n B a21 a22 ··· a2n C B C B . .. C @ . A am1 am2 ··· amn The following are matrices over Q (Remember Q is the field of rational numbers, i.e. numbers of a 1 2 the form b , where each a and b are integers. Like 2 , 7 ): 0 1 1 2 4 8 3 1 3 7 6 0:212121 ··· 0:5 1 2 3 ; @ 5 5 1 A ; 8 ; (5); 4 23 0:03 0:24 0:333 ··· 3 7 6 where in the last matrix the dots represents repeating digits, not the ignored elements. The following are matrices over R (Remember R is the field of all rational and irrational numbers, i.e. numbersp that can be expressed as finitely many integer digit and infinitely many decimal digits. 1 Like 3 , 2, π) 2 0 p 1 0 4 1 2 4 5 π 3 2 − 1 3π π ; ; 10 3 cos 1 @ A 1 7 @ 5 A e 5 2 6 10 The following are matrices over C (Remember C is the field of all complex numbers, i.e. numbers that can be expressed as a+bi where a and b are real numbers and i is the element satisfies i2 = −1) 0 1 3 2 1 0 3 + i 1 3 1 1 + i 2i 5 ; 6 2 8 ; π + 1 i 5 1 3 + πi 5 + 2i @ A @ 5 6 A 1 2 4 9 + 9i 1 5i Because we know rational numbers are included in real numbers, real numbers are included in complex numbers, so a matrix over Q is a matrix over R, and a matrix over R is a matrix over C 0 1 a11 a12 ··· a1n B a21 a22 ··· a2n C A = B C For a matrix B . .. C over F, @ . A am1 am2 ··· amn the rows of Matrix are called row vectors of Matrix, always denoted as rk, where the subindex are arranged by order: 1st row vector r1 = a11 a12 ··· a1n 2nd row vector r2 = a21 a22 ··· a2n ··· m’th row vector rm = am1 am2 ··· amn The columns of Matrix are called column vectors of Matrix, always denoted as ck, where the subindex are arranged by order: 0 1 0 1 a11 a12 B a21 C B a22 C c = B C c = B C 1st column vector 1 B . C 2nd column vector 2 B . C @ . A @ . A am1 am2 0 1 a1n B a2n C ··· c = B C n’th column vector n B . C @ . A amn p 0 3 1 2 1 For the matrix over R: A = @ 1 5 8 A. The row vectors are 1 3 5 2 p 1 r1 = 3 1 2 ; r2 = 1 5 8 ; r3 = 3 5 2 3 The column vectors are p 0 3 1 0 1 1 0 2 1 c1 = @ 1 A ; c2 = @ 5 A ; c3 = @ 8 A 1 3 5 2 The element aij, which located at i’th row and j’th column are called ij-entry of the Matrix. We frequently denote a matrix by simply writting A = (aij)1≤i≤m if the formula or rule of aij is explicitly given or clear. 1≤j≤n 0 1 3 5 1 Suppose (aij)1≤i≤3 = @ 2 1 9 A, What is a12? 1≤j≤3 7 0 8 Answer: a12 is the element getting by the first row and second column, so a12 = 3 What is the matrix (i + j)1≤i≤3? 1≤j≤3 0 2 3 4 1 Answer: @ 3 4 5 A 4 5 6 2 2 What is the matrix (i + j )1≤i≤1? 1≤j≤2 Answer: 2 5 Besids the notation (aij)1≤i≤m where aij is bunch of numbers. People can also write a matrix as a row 1≤j≤n vector of column vectors, or column vector of row vectors. Explicitely, row vector of column vectors is like A = c1 c2 ··· cn , this only make sense when each the size of each column vector ci are 0 1 r1 B r2 C A = B C equal. The column vector of row vectors is like B . C where each ri are row vectors, and only @ . A rm make sense when size of each of them are equal. r1 Suppose r1= 6 2 3 , r2= 4 9 , what is the matrix ? r2 Answer:This does not make sense because the size of r1 are not equal to the size of r2 r1 Suppose r1= 2 3 , r2= 1 5 , what is the matrix ? r2 4 2 3 Answer: 1 5 1 2 Suppose c = , c = , what is the matrix c c ? 1 0 2 9 1 2 1 2 Answer: 0 9 A matrix with m rows and n columns is called an m by n matrix, written as m × n. The pair of the number m × n is called the size of the matrix. What is the size of 1 3 ? Answer: 1 × 2 Two matrices A and B over F are called equal, written A=B, if they have the same size and each of the corresponing elements are equal. Thus the equality of two m × n matrices is equivalent to a system of mn equalities, each of the equality corresponds to pair of elements. x y 1 4 Solving the equation = z 5 2 5 Answer: as the 22-entry matches each other, this is possible for the equation make sense.