Lecture Notes for Math182: Algorithms Draft: Last Revised July 28, 2020
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LINEAR ALGEBRA METHODS in COMBINATORICS László Babai
LINEAR ALGEBRA METHODS IN COMBINATORICS L´aszl´oBabai and P´eterFrankl Version 2.1∗ March 2020 ||||| ∗ Slight update of Version 2, 1992. ||||||||||||||||||||||| 1 c L´aszl´oBabai and P´eterFrankl. 1988, 1992, 2020. Preface Due perhaps to a recognition of the wide applicability of their elementary concepts and techniques, both combinatorics and linear algebra have gained increased representation in college mathematics curricula in recent decades. The combinatorial nature of the determinant expansion (and the related difficulty in teaching it) may hint at the plausibility of some link between the two areas. A more profound connection, the use of determinants in combinatorial enumeration goes back at least to the work of Kirchhoff in the middle of the 19th century on counting spanning trees in an electrical network. It is much less known, however, that quite apart from the theory of determinants, the elements of the theory of linear spaces has found striking applications to the theory of families of finite sets. With a mere knowledge of the concept of linear independence, unexpected connections can be made between algebra and combinatorics, thus greatly enhancing the impact of each subject on the student's perception of beauty and sense of coherence in mathematics. If these adjectives seem inflated, the reader is kindly invited to open the first chapter of the book, read the first page to the point where the first result is stated (\No more than 32 clubs can be formed in Oddtown"), and try to prove it before reading on. (The effect would, of course, be magnified if the title of this volume did not give away where to look for clues.) What we have said so far may suggest that the best place to present this material is a mathematics enhancement program for motivated high school students. -
Sums and Products
1-27-2019 Sums and Products In this section, I’ll review the notation for sums and products. Addition and multiplication are binary operations: They operate on two numbers at a time. If you want to add or multiply more than two numbers, you need to group the numbers so that you’re only adding or multiplying two at once. Addition and multiplication are associative, so it should not matter how I group the numbers when I add or multiply. Now associativity holds for three numbers at a time; for instance, for addition, a +(b + c)=(a + b)+ c. But why does this work for (say) two ways of grouping a sum of 100 numbers? It turns out that, using induction, it’s possible to prove that any two ways of grouping a sum or product of n numbers, for n 3, give the same result. ≥ Taking this for granted, I can define the sum of a list of numbers a1,a2,...an recursively as follows. A sum of a single number is just the number: { } 1 ai = a1. i=1 X I know how to add two numbers: 2 ai = a1 + a2. i=1 X To add n numbers for n> 2, I add the first n 1 of them (which I assume I already know how to do), then add the nth: − n n−1 ai = ai + an. i=1 i=1 X X Note that this is a particular way of grouping the n summands: Make one group with the first n 1, and add it to the last term. -
Notes on Linear Algebra
Notes on linear algebra Darij Grinberg Tuesday 13th December, 2016 at 21:44 These notes are frozen in a (very) unfinished state. Currently, only the basics of matrix algebra have been completed (products, triangularity, row operations etc.). Contents 1. Preface3 1.1. Acknowledgments . .4 2. Introduction to matrices4 2.1. Matrices and entries . .4 2.2. The matrix builder notation . .6 2.3. Row and column vectors . .7 2.4. Transposes . .8 2.5. Addition, scaling and multiplication . .9 2.6. The matrix product rewritten . 13 2.7. Properties of matrix operations . 18 2.8. Non-properties of matrix operations . 21 2.9. (*) The summation sign, and a proof of (AB) C = A (BC) ....... 22 2.10. The zero matrix . 31 2.11. The identity matrix . 32 2.12. (*) Proof of AIn = A ............................ 33 2.13. Powers of a matrix . 35 2.14. (*) The summation sign for matrices . 36 2.15. (*) Application: Fibonacci numbers . 40 2.16. (*) What is a number? . 43 3. Gaussian elimination 51 3.1. Linear equations and matrices . 51 3.2. Inverse matrices . 53 3.3. More on transposes . 61 1 Notes on linear algebra (Tuesday 13th December, 2016, 21:44) page 2 3.4. Triangular matrices . 62 3.5. (*) Proof of Proposition 3.32 . 71 3.6. The standard matrix units Eu,v ...................... 75 3.7. (*) A bit more on the standard matrix units . 77 l 3.8. The l-addition matrices Au,v ....................... 84 3.9. (*) Some proofs about the l-addition matrices . 86 l 3.10. Unitriangular matrices are products of Au,v’s . -
Notes on Summations and Related Topics
Notes on summations and related topics James Aspnes December 13, 2010 1 Summations Summations are the discrete versions of integrals; given a sequence xa; xa+1; : : : ; xb, Pb its sum xa + xa+1 + ··· + xb is written as i=a xi: The large jagged symbol is a stretched-out version of a capital Greek letter sigma. The variable i is called the index of summation, a is the lower bound or lower limit, and b is the upper bound or upper limit. Mathematicians invented this notation centuries ago because they didn't have for loops; the intent is that you loop through all values of i from a to b (including both endpoints), summing up the body of the summation for each i. If b < a, then the sum is zero. For example, −5 X 2i sin i = 0: i3 i=0 This rule mostly shows up as an extreme case of a more general formula, e.g. n X n(n + 1) i = ; 2 i=1 which still works even when n = 0 or n = −1 (but not for n = −2). Summation notation is used both for laziness (it's more compact to write Pn i=0(2i + 1) than 1 + 3 + 5 + 7 + ··· + (2n + 1)) and precision (it's also more clear exactly what you mean). 1.1 Formal definition For finite sums, we can formally define the value by either of two recurrences: b ( X 0 if b < a f(i) = (1) f(a) + Pb f(i) otherwise. i=a i=a+1 b ( X 0 if b < a f(i) = (2) f(b) + Pb−1 f(i) otherwise. -
Short Course on Asymptotics Mathematics Summer REU Programs University of Illinois Summer 2015
Short Course on Asymptotics Mathematics Summer REU Programs University of Illinois Summer 2015 A.J. Hildebrand 7/21/2015 2 Short Course on Asymptotics A.J. Hildebrand Contents Notations and Conventions 5 1 Introduction: What is asymptotics? 7 1.1 Exact formulas versus approximations and estimates . 7 1.2 A sampler of applications of asymptotics . 8 2 Asymptotic notations 11 2.1 Big Oh . 11 2.2 Small oh and asymptotic equivalence . 14 2.3 Working with Big Oh estimates . 15 2.4 A case study: Comparing (1 + 1=n)n with e . 18 2.5 Remarks and extensions . 20 3 Applications to probability 23 3.1 Probabilities in the birthday problem. 23 3.2 Poisson approximation to the binomial distribution. 25 3.3 The center of the binomial distribution . 27 3.4 Normal approximation to the binomial distribution . 30 4 Integrals 35 4.1 The error function integral . 35 4.2 The logarithmic integral . 37 4.3 The Fresnel integrals . 39 5 Sums 43 5.1 Euler's summation formula . 43 5.2 Harmonic numbers . 46 5.3 Proof of Stirling's formula with unspecified constant . 48 3 4 CONTENTS Short Course on Asymptotics A.J. Hildebrand Notations and Conventions R the set of real numbers C the set of complex numbers N the set of positive integers x; y; t; : : : real numbers h; k; n; m; : : : integers (usually positive) [x] the greatest integer ≤ x (floor function) fxg the fractional part of x, i.e., x − [x] p; pi; q; qi;::: primes P summation over all positive integers ≤ x n≤x P summation over all nonnegative integers ≤ x (i.e., including 0≤n≤x n = 0) P summation over all primes ≤ x p≤x Convention for empty sums and products: An empty sum (i.e., one in which the summation condition is never satisfied) is defined as 0; an empty product is defined as 1. -
INTRODUCTION to the ARITHMETIC THEORY of QUADRATIC FORMS Contents 1. Review of Linear Algebra and Tensors 1 2. Quadratic Forms A
INTRODUCTION TO THE ARITHMETIC THEORY OF QUADRATIC FORMS SAM RASKIN Contents 1. Review of linear algebra and tensors 1 2. Quadratic forms and spaces 10 3. p-adic fields 25 4. The Hasse principle 44 References 57 1. Review of linear algebra and tensors 1.1. Linear algebra is assumed as a prerequisite to these notes. However, this section serves to review the language of abstract linear algebra (particularly tensor products) for use in the remainder of these notes. We assume the reader familiar with these notions, and therefore do not provide a complete and detailed treatment. In particular, we assume the reader knows what a homomorphism of algebraic objects is. 1.2. Rings and fields. Recall that a ring is a set A equipped with associative binary operations ` and ¨ called \addition" and \multiplication" respectively, such that pk; `q is an abelian group with identity element 0, such that multiplication distributes over addition, and such that there is a multiplicative identity 1. The ring A is said to be commutative if its multiplication is a commutative operation. Homomorphisms of rings are maps are always assumed to preserve 1. We say that a commutative ring A is a field if every non-zero element a P A admits a multiplica- tive inverse a´1. We usually denote fields by k (this is derived from the German “K¨orper"). Example 1.2.1. For example, the rational numbers Q, the real numbers R, and the complex numbers C are all fields. The integers Z form a commutative ring but not a field. -
Blastoff Language Reference Manual
BLAStoff Language Reference Manual Katon Luaces, Michael Jan, Jake Fisher, Jason Kao fknl2119, mj2886, jf3148, [email protected] Contents 1 Introduction 2 2 Lexical Conventions 2 2.1 Assignment . .2 2.1.1 Matrix Literal Definition . .2 2.1.2 Graph Definition . .3 2.1.3 Number Definition . .4 2.1.4 Generator Function Definition . .4 2.1.5 String Definition . .6 2.1.6 Integers vs. Floats . .6 2.2 Comments . .7 2.3 Functions . .7 2.4 If statements . .8 2.5 For/While Loops . .8 2.6 Operations . .9 2.6.1 Selection [] . .9 2.6.2 Matrix Multiplication * . 11 2.6.3 Convolution . 12 2.6.4 Element-wise Multiplication @ . 13 2.6.5 Element-wise Addition + . 13 2.6.6 Exponentiation ^ ...................... 14 2.6.7 Size jj ............................. 15 2.6.8 Vertical Concatenation : . 16 2.6.8.1 A note on horizontal concatenation . 16 2.6.9 Reduce Rows % . 16 2.6.9.1 A note on matrices where m =0......... 17 2.6.9.2 A note on reduce columns . 18 2.6.10 Assignment operators *=, =, @=, +=, ^=, := . 18 2.6.11 Comparisons ==; 6=>; ≥; <; ≤ ................ 18 2.6.12 Semiring redefinition # . 19 2.6.12.1 A note on matrices where m = 0, again . 20 2.6.13 Logical Negation ! . 20 2.7 Precedence . 21 2.8 Keywords . 21 3 More Language Details 21 3.1 Memory . 21 3.2 Scope . 22 3.3 Printing . 22 1 4 Sample Code 23 4.1 Some Standard Library Functions . 23 4.1.1 One . 23 4.1.2 Horizontal Concatenation . -
Math 127: Induction
Math 127: Induction Mary Radcliffe 1 Induction Fundamentals The moment we've all been waiting for: a full treatment of proof by induction! Before we get into the technique, here, let us first understand what kinds of propositions we wish to treat using induction. As a first understanding, we will consider propositions that take the basic form: 8n 2 N; p(n) (1) where p(n) is a proposition about n. For example, suppose we wish to prove something like the following: n(n + 1) 8n 2 ; 1 + 2 + 3 + ··· + n = N 2 Using the techniques we previously developed in the notes about logic and proof, you might find yourself at something of a loss. While we are able to prove this proposition directly, let's imagine we don't see a way forward with direct proof. Contradiction and contrapositive seem pretty infeasible here, since assuming that two numbers are not equal doesn't really give us much to work with. So we'd like to develop a different path. But, ah! We remember something useful: the Induction Axiom from the Peano Axioms! In case you've forgotten, let's restate it here: If S is a set of natural numbers, having the property that 1 2 S and that if a 2 S, then (2) the successor a+ of a is also in S, then S = N. To apply that to our current situation, we can look to where we wish to have all of N: we wish that every number in N makes the proposition true. So we could imagine that we have a set S, where we define S by saying that n 2 S if and only if p(n) is true.