Finite Mathematics. Linear Equations (Chapter 4) Optimization (Chapter 5) Finite Probability (Chapter 6) Markov Chains (Chapter 9) Finite Mathematics

Transcription

1 Finite Mathematics Linear Equations (Chapter ) Optimization (Chapter ) Finite Probability (Chapter 6) Markov Chains (Chapter 9) Finite Mathematics Linear Algebra Probability Linear Equations Simple Method Markov Chains

2 Systems of Linear Equations Matrices Equations in two variables Augmented Matrices Gauss-Jordan Algorithm Matri Operations Inverses of Square Matrices Matri Equations Leontief Input-Output Analysis. Systems of Linear Equations in Two Variables (eview) Systems in two variables Graphing Substitution Elimination by Addition Applications

3 Eample You are buying fruit. If oranges and apple cost $. and if orange and apples cost $., what is the price of each? Let be the price of an orange and y be the price of an apple. Then we have y. y. Find and y! Given the linear system abyh cdyk where a,b,c,d,h,and k are real constants, a pair of numbers and yy is a solution of this system if each equation is satisfied by that pair. all solutions solution set solve find solution set

4 Given the linear system a by h c dy k where a,b,c,d,h, and k are real constants, a pair of numbers and yy is a solution of this system if each equation is satisfied by that pair. All solutions solution set solve find solution set Solution by graphing Each linear equation in variables is a line, the solution of a system of two such equations is the point where these lines intersect (if they do). y

5 Slope m rise run ymb y (,y ) (,y ) y run - ise y -y b Slope intercept form y m b Point slope form y y m( ) Two point form y y y y ( )

6 Possible solution sets (a) - y y, y (b) y y - 8 none (c) y y 8 -y, y arbitrary Basic Terms consistent one or more solutions eist inconsistent no solution eists independent consistent and eactly one solution eists dependent consistent and multiple solutions

7 Solution By Substitution Solve using substitution - y 7 y First solve one of the equations for y. -y7- or y-7 Now substitute this into the other equation: (-7) or 9- Solve for and substitute back to get y: or and y()-7 or y- Theorem : Possible solutions to a linear system The linear system abyh cdyk must have (A) eactly one solution (consistent & independent) O (B) no solution (inconsistent) O (C) infinitely many solutions (consistent & dependent) there are NO other possibilities

8 Elimination By Addition Transformation to a simpler equivalent system Theorem : Operations that produce equivalent systems: (i) two equations are interchanged (ii) an equation is multiplied by a nonzero constant (iii) a constant multiple of one equation is added to another equation Solve - y -8 y Add both equations to get 7-7 which gives - Now use this in either equation (-) - y -8 -y -6 y - The solution is y or (-,)

9 SUPPLY AND DEMAND The supply and demand equations are linear, that is p aq b where pprice, qquantity, a,b unknown. Supply data: at a price of $. per bushel, there are 8 bushels for sale, if the price drops to $., only bushels are for sale. Find the supply equation:. a 8 b. a b find a,b Substitution: solve the first equation for b b. - 8 a put this into second equation. a. - 8 a.. - a -. - a. a put this back into b. - 8 (.).6 SUPPLY EQUATION: p. q.6

10 . Systems of Linear Equations and Matrices Matrices Solving systems of equations Augmented matrices Summary MATICES A Matri is a rectangular array of numbers written within brackets: A. B

11 Each number is called an element of the matri If there are m rows and n columns then we say the matri has dimension m n or that it is an m n matri. A is a matri and Ba matri If a matri has n rows and n columns we say it is a square matri, if it has only one row it is a row matri, and if it has only one column, then we call it a column matri. A is a square matri. C [ -. ] D C is a column matri of dimension D is a row matri of dimension

12 The position of an element is the row number and column number of the element: A. The element in position (,) is, in position (,) is -. We write a i,j for the element in the i-th row and j-th column. Augmented matrices

13 Two systems of equations are equivalent if they have the same solution set. Two augmented matrices A and B are row equivalent, written A ~ B, if the corresponding systems of equations are equivalent. THEOEM Operations producing row equivalent matrices (A) Two rows are interchanged i j (B) A row is multiplied by a nonzero constant k i i (C) A constant multiple of one row is added to another row k j i i

14 Solving linear systems using augmented matrices Starting with - 6 we want to end up with - or in terms of augmented matrices So we are trying to transform the initial augmented matri into the form c c cm c m

15 Problem Translating back to an equation gives: check : or ) ( ) ( ) (

16 or inconsistent SUMMAY Unique solution consistent and independent Infinitely many consistent and dependent No solution inconsistent p n m n m n m Where m,n,p are real numbers and p

17 . Gauss - Jordan Elimination EDUCED MATICES SOLVING SYSTEMS BY GAUSS - JODAN ELIMINATION APPLICATION Possible Solutions to Linear Systems UNIQUE SOLUTION (consistent and independent) NO SOLUTION (inconsistent) INFINITELY MANY SOLUTIONS (consistent and dependent)

18 EDUCED MATIX Form Form Form m m n m n n p m,n,p real numbers,p EDUCED MATIX A matri is in reduced form if: ) Each row consisting entirely of zeros is below any row having at least one nonzero element ) The leftmost nonzero element in each row is ) All other elements in the column containing the leftmost of a given row are zeros ) The leftmost in any row is to the right of the leftmost in the row above

19 9 9 8 Solving systems by Gauss-Jordan Elimination Gauss-Jordan Elimination is the systematic process of getting a matri into reduced form. Move rows of zeros to bottom Get in top left corner Get s below Get in second row second column Get s below and above this one Keep going (third row, third column etc.) Stop when in reduced form

20 # reduced

21 Steps for Gauss-Jordan Elimination Step Find leftmost nonzero column, get to top Step Get zeros above and below leading Step epeat Step on submatri formed by (mentally) deleting row used in step and all rows above Step epeat Step with entire matri, including the mentally deleted rows 9 9 9

22

23 6: A commuter airline wants to purchase a fleet of airplanes with a combined carrying capacity of 96 passengers. Three available plane types carry 8,, and passengers, resp. number of planes carrying 8 passengers number of planes carrying passengers number of planes carrying passengers

24 is a free variable, it does not depend on the others let t and we have to choose t such that,, are nonnegative integers (planes do not come in halves ) t implies t >., so t,, 7 t implies t < 7., so t 67,,, so we have (,,),(,,),(8,6,6),(,,7) We have feasible scenarios: ) planes carrying 8 passengers, carrying passengers, carrying passengers ) planes carrying 8 passengers, carrying passengers, carrying passengers ) 8 planes carrying 8 passengers, 6 carrying passengers, 6 carrying passengers ) planes carrying 8 passengers, carrying passengers, 7 carrying passengers

25 . MATICES: Basic Operations ADDITION and SUBTACTION PODUCT OF A NUMBE k AND A MATIX M MATIX PODUCT WHEN AE TWO MATICES EQUAL? ECALL: Dimension of a matri number of rows times number of columns m n matri has m rows, n columns 8 this is a matri

26 Only matrices of the same dimension can be added to each other! ( ) 6 Adding an m n matri to an m n matri gives a m n matri as a result where each entry is the sum of the entries of the two matrices being added. 9 NOT POSSIBLE!!!

27 NUMBE TIMES A MATIX ( ) 7 7 Any matri can be multiplied by any number zero matri of dimension ( rows, columns)

28 MATIX PODUCT Product of a n row matri and a n column matri: b b bn [ a a an] [ ab ab anbn] ( n) times (n ) ( ) matri Labor cost: Chair requires hours in assembly, hour in finishing department. Each hour in assembly costs $, each hour in finishing costs $, what is the production cost of one chair? [ ] [ ] [ 7] Each chair costs $7 to produce.

29 Matri Product The matri product of a (m p) A and a (p n) B matri is a (m n) matri, whose element in the i-th row and j-th column is the real number which results when multiplying the i-th row of A and the j-th column of B. [ ] [ ] [ ] [ ]

30 To multiply a matri A by a matri B, that is to form the product AB, the number of columns of A (second inde) has to be the same as the number of rows of B (first inde) NOT POSSIBLE!!! Matri Multiplication is NOT commutative! even the same dimension not

31 AB BA even if both products are defined BA even if both products are defined BA even if both products are defined BA even if both products are defined and of the same size! and of the same size! and of the same size! and of the same size! Identity matri AA AA and IA AI i h g f e d c b a i h g f e d c b a i h g f e d c b a d c b a d c b a d c b a

32 For real numbers: a b implies a or b More potential pitfalls: AB zero matri does NOT imply that either A zero matri or B zero matri. INVESE OF A SQUAE MATIX Identity Matri Inverse of a square Matri Applications to Cryptography Square means same number of rows and columns

33 Some facts about multiplication For non zero real numbers, we have an identity and inverses: and Identity element for multiplication The following will work for ( ) and ( ) matrices.

34 Note for real numbers we have or - For matrices we can have INVESE OF A SQUAE MATIX For real numbers a a a a For matrices a c b d What should a,b,c,d be?

35 Do the matri product work with the equations: a c b d a c b d a c b d a c b d Now solve these two systems simultaneously! Finding the inverse of a matri Using augmented matrices for the two systems gives: The left-hand sides are equal! So combine:

36 Now solve this double system so To find inverse of square matri M: augment M by identity matri I (same number of rows and columns) perform row operations (create identity to left of vertical line) M - appears on right

37 THEOEM. If [M I] is transformed by row operations into [I B], then the resulting matri B is M -. However, if we obtain s in one or more rows to the left of the vertical line, then M - does not eist. does not have an inverse. 9 also does not have an inverse We can not create a in the last column.

38 No inverse eists if ) M has a row of all zeros ) M has a column of all zeros ) Case ) or ) occurs during row operations does not have an inverse -

39 For matrices there is an easy way a b c d ad bc d c b a No simple formula for larger matrices! Cryptography (encoding/decoding) A B C D Y Z _ 6 7 WE DO NOT LIKE POP QUIZZES

40 To decode we need the inverse of the encoding matri: Matri Equations Systems of Linear Equations Matri Equations Systems of Linear Equations Applications

41 Basic Properties of Matri Arithmetic Addition Properties: Associative (AB)CA(BC) Commutative ABBA Additive Identity AA Additive Inverse A(-A) Multiplication Properties: Associative (A B) CA (B C) Multiplicative Identity A I I AA IF A is a SQUAE matri and A - eists: Multiplicative Inverse A A - I In general matri multiplication is NOT commutative: A B B A

42 Combined Properties Left Distributive A(BC) (AB) (AC) AB AC ight Distributive (BC)A BA CA Equality Properties Addition Left Multiplication ight Multiplication If A B then AC BC If A B then CA CB If AB then AC BC Solving Equations For eal Numbers: a,b,c, known; unknown abc add -b to both sides ab(-b)c(-b) ac-b multiply by a - from the left a - aa - (c-b) a - (c-b)

43 Solving Matri Equations Find,y,z in the matri equation below K X A are given,, where m l k m l k z y Solution: Find A - and multiply both sides from the left by A - : K A X K A A X A Find the inverse - A

44 Solution of the Matri Equation m l k m l k m l k z y m l k z y K A X - Why Use Matri Equations? Need to calculate A - only once If we have many different scenarios (k,l,m) then there is no need to run through Gauss-Jordan Elimination many times If we have only one case, then there is no advantage.

45 Problem 6 Production of two models A,B. If $, a week is allowed for labor and material, how man of each model should be produced each week to use up all the available money? Cost: Model Labor Cost Material Cost A $ $ B $ $ Weekly allocation: Labor $8 $7 $7 Material $ $ $8 Solution: Let be the number of units of model A produced per week and y the number of units of model B. Week Week Week y8 y y7 y y7 y8 l y m y l m

46 Finding the inverse.... Answer to the question y y y In week we need to produce 6 items of model A, in week each of models A and B and in week we need to produce items of model A and of model B.

47 .7 Leontief Input-Output Analysis INTODUCTION TWO-INDUSTY MODEL THEE-INDUSTY MODEL WASSILY LEONTIEF NOBEL PICE IN ECONOMICS 97 sectors of American economy study of interaction long term economic planning

48 TWO-INDUSTY MODEL Elec. ELECTICITY ELECTIC COMPANY WATE WATE COMPANY Water ELECTICITY WATE CONSUME DEMAND SETTING UP THE MODEL All calculations in dollar value (same units for electricity and water) Per dollar production (output) needed electricity (input) needed water (input) consumer demand electric company $. $. $ million water company $. $. $ 8 million

49 Basic Input-Output Model Given the internal demand for each industry s output, determine output levels for the various industries that will meet a given final (outside) level of demand. How much electricity/water do we need to produce to satisfy internal and eternal demands? Linear equations Suppose E produces eactly $ million of electricity and W eactly $ 8 million of water.().(8) $. (internal demand for E) electricity for consumer $-$.$6.8.().(8) $. (internal demand for W) water left for consumer $8-$.$.6 So increase water and electricity production, but by HOW MUCH?

50 Technology matri Input E W E Input from E to produce $ of electricity Input from W to produce $ of electricity O ut p u t W Input from E to produce $ of water Input from W to produce $ of water Better linear equations Let total output from electric company total output from water company d eternal demand for electricity d eternal demand for water.. Internal demand for electricity.. Internal demand for water.. d.. d

51 Total Output Matri Equation Internal Demand Eternal Demand.. d.. d Matri Equation X MX D where D d d X.. M.. Solution of the matri equation XMXD X-MXD IX-MXD (I-M)XD X(I-M) - D we want X so put all X on the left Note that XIX Factor out X to right If (I-M) is invertible, then is the solution.

52 Solution of electric/water problem.. X X (-.)(-.) X Thus and 7 and since we used millions of dollars as units, we get that we need to produce $ million worth of electricity and $7 million worth of water.

53 Three-industry model electricity (E) #8 gas (G) oil (O) $ of E requires $. of E, $. of G, $. of O $ of G requires $. of E, $. of G, $. of O $ of O requires $. of E, $. of G, $. of O Technology matri Output E G O E... Input G... O... Solution Demand: $ billion for E, $ billion for G and $ billion for O. Let total E, total G, total O We thus need to produce $ billion of E, $7 billion of G and $ billion of O.