When Life is Linear From Computer Graphics to Bracket010gy

When Life is Linear From Computer Graphics t0 Bracket010gy Tim Chartier Da ⅵ ホ 0 〃 Co 〃 ビ g ビ P ″ わ 〃 4 〃 イ D な 町 e イ The Mathematical Association Of AmerIca

一 、 こ 蠱 物 ー き ハ ゞ ゞ ′ ヾ ゞ 日 S LINEAR WHEN FROM COMPUTER GRAPHICS T0 BRACKETOLOGY TIM CHARTIER MATHEMATICAL A 0 ロ A 引 ON OF AMERICA ANNil_l LAX NEW MAT.-IEMAT!CAI II±RARY

36. Game Theory and Strategy の , P ん 〃 〃 D. & 加 〃 , 五 37. Episodes in Nineteenth and Twentieth Century Euclidean Geometry RO 〃 0 わ 夜 ・ g 夜 ・ 38. The Contest Problem Book V American High School Mathematics Examinations and American lnvitational Mathematics Examinations 円 83 一 円 88. Compiled and augmented の , G ビ 0 ′ ・ g ビ B ビ に 〃 4 〃 イ & ビ 〃 / に 〃 B. M 砒 〃 ・ げ 39. Over and Over Again の , Gengzhe Cha 〃 g 4 〃 d 〃 10 川 毖 立 施 夜 ・ g 40. The Contest Problem BOOk VI American High School Mathematics Examinations 円 89 一 円 94. CompiIed and augmented ん ビ 0. / 立 / Ⅲ 施 ド 41. The Geometry of Numbers の , C. D. 0 / ホ , 〃 〃 el / ん , イ GiuIiana 2 42. Hungarian Problem BOOk Ⅱ し Based on the Eötvös Competitions 円 29 一 円 43 , ″ ハ / 4 d の , 月 〃 ん ⅲ 43. Mathematical Miniatures & 哲 / の , S の , c / に リ 4 〃 d 7 ・ ⅲ 信 4 〃 市 ℃ ビ ″ 44. Geometric Transformations IV /. M. g / 0 ′ 〃 , / ra d の , 月 . S / ⅲ に 夜 ・ 45. When Life is Linear: from computer graphics to bracketology 7 カ 〃 C ん ah ⅵ 夜 ・ 0 励 夜 ・ titles ⅲ 〃 ′ ℃ 〃 ra 行 0 〃 . MAA Service Center P.. O. Box 91 Ⅱ 2 Washington, DC 20090-1 Ⅱ 2 ト 800-3 引 - 1 MAA FAX: 1-240-396-5647

Preface the 1999 film The ん ″ ・ ⅸ , there is the following interchange between Neo and Trmity. Neo: Trinity: Ⅲ カ 4 ー パ ビ イ ″ ・ ? The の ハ ル 夜 ・ 0 夜 ℃ , Neo, 4 〃 d ″ 3 / 00 石 〃 g As an applied mathematlcian specializing in linear algebra, I see many applications Of linear systems in the WO 日 d. From computer graphics tO data analytics, linear algebra is useful and powerful. SO, t0 me, a matnx, connected tO an application, is indeed out there—waiting, ⅲ a certain sense, tO be formed. When you find it, you can create compelling computer graphics or gam insight on real world phenomenon. This book is meant to engage high schoolers through professors of mathematlcs in applications 0f linear algebra. The b00k teaches enough linear algebra tO dive intO applications. げ someone wanted tO know more, a Goog に search will turn up a lOt Of information. げ more information is desired, a course or textbook in linear algebra would be the next step. This book can either serve as a fun way tO step intO linear algebra or as a supplementary resource for a class. My hope that this b00k will ignite ideas and fuel innovation in the reader. From trying Other datasets for data analytics tO vmations on the ideas ln computer graphics, there is plenty Of room for your personal discovery. SO, this b00k is a stepping stone, intended t0 sprmgboard you intO exploring your own ideas. This book is connected t0 a Massive Open Online Course (also known as a MOOC) that willlaunch in February 0f2015 through Davidson C011ege

XII When Life is Linear and edX. The bOOk and online course are not dependent on each other. Participants 0f the MOOC d0 not need this book, and readers 0f this b00k d0 not need t0 participate ⅲ the MOOC. The expenences are distinct, but for those interested, I believe the bOOk and online course will complement each other. One benefit of the MOOC is a collection of online resources being developed. Many 0f the ideas in this b00k require a computer and either mathematical software or a computer program tO calculate the underlying mathematics. From graphics that have thousands Of pixels tO datasets that have hundreds or thousands Of entnes, computing can be necessary even for ideas that can be easily demonstrated by hand on small datasets. ロ 00k forward to seeing the ideas ofthis book shared, explored, and extended with the online communlty. The book has one more goal. A few years ago, a group of students gathered for a recreational game 0f Math Jeopardy. One 0fthe subJect areas was linear algebra, which turned out t0 be the hardest for the students. As the students struggled tO recall the content necessary tO answer the questions, one student 」 0ked, 、 、 Well, at least this stuff isn't useful. Maybe that's why I can't remember it. " Even looking at the table ofcontents ofthis book, you'll see why I see linear algebra as very useful. I tend t0 believe that at any second in any day linear algebra is being used somewhere in a real world application. My hope is that after reading this b00k more people will say, 。 1 really need t0 remember these ideas. They are SO useful!" If that person IS you, maybe you'd begin sharmg the many ideas you created and explored as you read this bOOk コ wish you well as you discover matrlces awaiting tO be formed.

MATHEMATICAL ASSOCIATION OF AMERICA ANNELI LAX NEW MATHEMATICAL L8RARY 日 S LINEAR WHEN FROM COMPUTER GRAPHICS TO BRACKETOLOGY TIM CHARTIER 「 om simulatlng complex phenomenon on supercomputers t0 storing the coordinates needed in modern 30 p ⅱ 而 叫 data is a huge and growing part 0f ou 「 wo . A m 可 0 「 to to manipulate and study this data islinear algebra. This book introduces concepts 0f matrl)( algeb 「 a with an emphasis on applicaüon. partlcularly in ds 0f comp 廰 「 graphics and data mining. Readers willlearn t0 make an image transparent compress an image and 「 0 ね te a 30 wireframe model.ln data mining, 「 eaders will use linear algebra t0 「 ead zip codes on envelopes and encrypt sensltlve information. The b00k details methods behind web search. utilized by such companies as Goog 厄 . and a 垣 0 「 曲 ms f0 「 sports ranklng which have been applied t0 creating brackets f0 「 March Madness and predict outcomes in FIFA World Cup S000e に The b00k can serve as own resource 0 「 tO supplement a course on linear algebra ISBN: 978-0-88385-649-9 乃 朝 4 ー 催 as 盟 な 例 ー 加 pe け 覊 カ 川 例 れ 04 ″ 4 e 房 4 c 側 な e. 方 り , ″ 0 た な イ 例 り 4 加 〃 ca 加 郷 , 覊 as co 襯 み 添 , c ワ og ′ り , 齠 切 e / 0 $ ra 加 g 側 イ 切 加 & e 加 g ル な み 側 々 加 催 側 co, ″ 4 房 4. Harvey Mudd CoIIege —ARTHUR N 」 AM . as 石 襯 朝 4 市 催 as $ , み 舫 み 例 g 加 g $ り 4 襯 0 翔 ん 4 り 坊 郷 研 4 ″ 4 4 働 ac 0 Ⅷ 〃 na り 齠 肥 4 / ″ ル み 側 々 刎 阨 c 郷 the ″ 側 イ 20 催 ー ん 舫 e ″ 4 e 房 4 側 イ 川 ル e ゆ 刎 加 ノ 可 雇 President and Professor, Southwestern University —EDWARD B. BURGER. / 切 イ 盟 加 c み 4 as / 4 “ ″ 4 愈 〃 ev 催 り が 04 / ″ 4 e 房 4 な / 加 み 催 ワ 阨 ゆ が / ca ″ 0 郷 , 0 〃 ゆ が / ca 〃 0 郷 4 尾 co 〃 イ 朝 4 催 な 4 尹 加 〃 g 側 geas ″ な イ 例 り 4 / 雇 ゆ が / ca 加 舫 4 ル 加 剏 4 ″ / 4 0 力 な 側 市 例 化 ア ho 襯 G 側 g な 4 尾 ん sa 加 ge. 側 e 加 g , ね 0 ホ 加 g $ イ ( 襯 ァ vo の co 襯 g ′ ゆ 加 —TONY DEROSE, Pixar Animation Studios

6 When Life is Linear 1 っ ) に ツ 只 ′ 広 ) 1 0 5 5 5 1 ^ リ 5 ^ 0 :.D 1 3 5 8 3 1 3 8 8 8 1 3 8 3 っ ) 1 0 8 3 1 3 8 0 ^ ソ 1 0 5 3 1 1 0 8 0 : 1 1 8 3 0 1 1 に ) 只 ) O 1 0 8 8 3 1 0 0 0 ) 3 Figu re 21. A well-known image created with a prlnter and numbers, but to much less satisfying effect than with a brush and paint. is a submatrix, which you'll see in Figure 2.1. SO, where is 図 ? lt's part 0f Mona Lisa's nght eye. Here lies an important lesson in linear algebra. Sometimes, we need an entire matr1X, even if it is large, tO get insight about the underlying application. Other times, we Ok at submatnces for our insight. At times, the hardest part is determining what part 0f information is most helpful. 2.1 Sub Swapping AS we gain more tOOls tO mathematically manipulate matnces, we'lllook at applications such as finding a group of high school fnends among a larger number of Facebook frlends. We'll identify such a group with submatrlces. The tools that we develop will help with such identification. WhiIe such mathematical methods are helpful, we can d0 interesting things simply with submatrices, particularly in computer graphics.

54 When Life is Linear SO, now we have a method ofsolving Ax = b. We can use elementary row operations tO compute such a solution and know hOW tO apply this t0 encrypting data. When we solve a linear system like 月 x = b we've computed an exact solution tO the linear equations. ln the next chapter, we see what can be done when there isn't a solution. ln such cases, we'll find a x that at least in a mathematical sense, comes close tO approximating a solution. SUPERCOMPUTING AND LINEAR SYSTEMS The U. S. nationallaboratones house many of the world's fastest high performance computers. One Of the fastest computers ⅲ the world (in fact the fastest in June 2010 ) is Jaguar (pictured above) housed at Oak Ridge National Laboratory that can perform 2.3 quadrlllion floating point operations per second. Such computers are commonly used tO solve large linear systems, sometimes containing billions Of vanables, WhiCh neces- sitates the use Of large-scale parallel computing. These massive linear systems Often result from differential equations that descrlbe complex physical phenomena. lmage co Ⅲ セ リ 研 04 ん Ridge ル 4 加 〃 記 ん わ ora 駻

52 When Life is Linear you get a recognizable message. That's especially fast t0 try on a computer. Often, we encrypt messages so they can't be easily broken. So, let's try a more complicated cipher. Rather than adding a vector tO another vector, we'll multiply a vector by a matrIX tO get the encrypted vector. This time we'll break our vector into groups 0f 3 before encoding. For this example, we'll multiply by the matnx 0 1 1 4 3 4 This will result in the sequence of numbers for the encrypted message. Let's agam encrypt the word LINEAR which, as we found, corresponds to the vector い 1 8 13 4 0 1 7 ] We break this vector intO vectors Of length 3 and treat each as a column vector. First, we compute 4 which modulo 26 is [ 21 21 16 ] You can verlfy that [ 4 0 17 ] becomes [ 24 17 6 ] . Combining the encoded vectors, we form the vector [ 4 0 17 24 17 6 ] which corresponds to the encrypted message: VVQYRG How d0 you decode this message? lt needs to be a quick and easy process for the receiver. First, ofcourse, we would transfer from the letters to find the first encrypted vector [ 21 21 16 ] TO decrypt, we want to find the vector x such that 1 8 00 っ 0 -1 00 3 0 4 ー 109 120 21 0 21 4 4 16 N0t surpnsingly, this is precisely the topic of this ch 叩 ter. EIementary row operations could be used within Gaussian elimination tO solve this linear system and find x ー い 7 216 一 円 5 ] RepIacing each ofthese entrles with the remainder when divided by 26 gives us い 1 8 13 ] っ 0 1