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Linear MathematicsOverviewLinear mathematics deals with linear equations. An equation is "linear" if it consists of a sum of variables or unknowns, each of which is multiplied by some number or constant (examples will be given below). Many real-world problems in physics, engineering, business, chemistry, biology, and other fields are described by linear equations. Computers are use to solve linear equations in groups or "systems," making possible many kinds of medical and scientific imaging, realistic video games, cheaper design of cars and other products, and the more efficient management of money. Fundamental Mathematical Concepts and TermsLinear equations are called "linear" (line-like) because the simplest kind of linear equation—one having two variables—describes a straight line. For example, the equation 2x0 + 3x1 = 4 describes the straight line depicted in Figure 1. Here x0 and x1 are "variables," meaning that they stand for any numbers we like; the small 0 and 1 are labels to tell them apart by. For each x0 we choose, there is one and only one x1 that makes 2x0 + 3x1 = 4 true. For example, if we set x0 equal to 0, then x1 must be 4/3 because: We can also use letters to stand for the fixed numbers that multiply x0 and x1. If we replace 2 and 3 in 2x0 + 3x1 = 4 with the symbols a1 and a2, and replace 4 with b—where these new letters can stand for any fixed numbers we like—we get a general-purpose linear equation in two variables: a1x1 + a2x2 = b. We can extend this to as many multipliers (also known as "coefficients") and variables as we like. The equation is still called a "linear" equation no matter how many variables we add. Here is the form of linear equation involving 3 variables and 3 coefficients: We have already seen how a two-variable linear equation describes a line. A three-variable linear equation
describes a plane, a set of points resembling a stiff sheet of paper tilted in space. In general, a linear equation containing n variables and n coefficients looks like this: The three dots in the middle of the equation stand for all the terms between the fourth term and the nth term that we don't want to bother to write down. In real-world applications, linear equations containing dozens or even millions of terms are common. Linear equations can be combined into groups or systems. A system of linear equations is a group of two or more equations that involve the same variables. The following is a system of two linear equations involving the two variables, x0 and x1: The "solution" of a system of linear equations is that set of numbers which, if plugged in for the variables, makes every equation in the system true at the same time. In this example, the solution of the system is x0 = 12/5, x1 = −4/15. This solution is unique; that is, each equation considered by itself is true for many values of x0 and x1, but only at x0 = 12/5, x1 = −4/15 are both equations true. If you graph the two equations in this system as lines on paper, the solution of the system will be the point where the two lines intersect. Every system of equations has a single, unique solution (like this system), or no solutions, or an infinite number of solutions. Among
systems that consist of two lines, like the ones that we've just been looking at, those that have a single, unique solution are lines that intersect (one point in common); those that have no solutions consist of parallel lines (no points in common); and those with an infinite number of solutions consist of two equations for the exact same line (all points in common). Systems of equations can also be written as matrix equations. A matrix is a rectangular array of numbers or variables with square brackets around it. It is named according to how high and how wide it is. For example, the matrix shown in Figure 2 is a 2 × 2 ("two by two") matrix because it is 2 entries tall and 2 entries wide. The matrix depicted on Figure 3 is a 2 × 3 ("two by three") matrix because it is 3 entries tall and 2 entries wide. A matrix can be added to, subtracted from, or multiplied by other matrices. It can also be multiplied by numbers, variables, and vectors, which are special matrices only 1 entry wide. Vectors containing three entries, as depicted in Figure 4, are particularly useful in science, engineering, and computer animation because each three-entry vector can specify a point, force, velocity, or acceleration in three-dimensional space.
The system of two equations shown earlier can be written as a 2 × 2 matrix multiplied by a vector and set equal to a second vector. That is, When there are only two or three variables in a system of equations, as in this example, there is no advantage in using the matrix form. But when systems involve many variables, as in most real-life applications, the matrix form is more efficient and revealing. Computers are well-suited to calculating with matrices, and are often used to solve systems whose matrices contain millions of entries. In creating medical images of the inside of the body, searching for oil reserves, predicting global climate change, designing new drug molecules, maximizing profits, and many other applications, the solution of large matrix equations by computer is key. Because the solution of systems of linear equations is so important in our high-technology society, most of the examples of linear math given below involve the solution of such systems. Real-life ApplicationsEARTHQUAKE PREDICTIONScience foresees no way of preventing earthquakes, which occur when whole sections of the Earth's crust, many miles across and weighing billions of tons, slip past east other. These forces are too great to control. However, knowing when and where earthquakes are likely to happen, and how strong they are going to be, would make better preparedness possible and reduce the loss in lives and money caused by major quakes. Linear equationsLinear equations that involve two variables, such as 2x + 3y = 4, describe straight lines. That is, if you graph any of the x, y pairs that satisfy the equation, you will find that they all lie on the same line on the paper—and, likewise, that every point on that line satisfies the equation. A linear equation that involves not two but three variables, such as 2x + 3y + 7z = 4, graphs a plane in three-dimensional space. Linear equations appear everywhere in science, technology, and business. If you are selling sneakers at x dollars of profit a pair, you know that if you sell 20 pairs of sneakers you will make double the money than if you sell 10 pairs, namely, 20x dollars rather than 10x dollars. Here the relationship of pairs sold to total profit is described by a linear equation: number of pairs sold (call it a) times profit per pair (x) equals total profit (p), ax = p. Anyone running a lemonade stand knows this much linear math by instinct. But not everything in real life is linear. For example, you might make more profit per pair of sneakers if you sell a million pairs than if you sell only a hundred. In this case, the equation describing your total profit in terms of sales will not be a linear equation. Nor are all linear equations as bare-bones as ax = p. If you are selling two types of sneaker, one of which makes x1 dollars of profit per pair while the other makes x2 dollars, a different linear equation arises. Say you sell a1 pairs of the first type of sneaker and a2 of the second type. Then your total profit p is given by the sum of the profits from each type: a1x1 + a2x2 = p. This is also a linear equation, but it involves two variables. In real business and industry, equations of this type involving variables are common. It is not yet possible to predict most earthquakes, but with the help of large systems of linear equations solved by computers, scientists are making rapid progress. The basic method, called "finite element modeling," is a common one in manufacturing, science, medical imaging, and other fields today. In finite element modeling, a mathematical model or image of an object or volume of space is built up using either triangles (for flat models) or tetrahedra (four-pointed pyramids, for three-dimensional models). The triangles or tetrahedra are called "elements" and fit together into a web or network called a "mesh." One or more separate variables (like x0, x1, and so forth used above) are assigned to each element, and linear equations involving these variables are written so that they approximate the laws of physics that apply in that area of space. For earthquake prediction, meshes containing 100 million tetrahedra or more are created that represent parts of Earth's crust containing earthquake faults. Equations are constructed using these meshes that describe how shock waves move through the rock and soil. Supercomputers are then used to solve the resulting systems of millions of linear equations; the solution shows what an earthquake will look like. Linear InequalitiesEquations express equalities, such as 1 + 2 = 3. We can also write inequalities, expressions that say that one thing is less (or greater) than some other. Four signs are used to express inequality: < (less than), ≤ (less than or equal to), > (greater than), and ≥ (greater than or equal to). For example, the expression a > b reads "a is greater than b." The expression c ≤ a reads "c is less than or equal to a." A linear inequality is a linear equation with its equals sign replaced by an inequality sign. The linear equation, x + y = 2, for example, can become the linear inequality x + y ≤ 2. The linear equation x + y = 2 describes a straight line—and that's why it's "linear" (See Figure A.)
The linear inequality x + y ≤ 2 also describes a set of points. The line x + y = 2 is part of that set because the "less than or equal to" sign includes an equals sign.
The inequality is also true for all points below the line, namely the gray area in Figure B. Linear inequalities arise often in real life. Consider a factory that can make two kinds of computer chip, Chip One and Chip Two, but cannot make both at the same time. The time needed to make a batch of Chip One is 5 minutes, so the time to make c1 batches of Chip One is 5c1 minutes. The time needed to make a batch of Chip Two is 10 minutes, so the time to make c2 batches of Chip Two is 10c2 minutes. But there are only 1,440 minutes in a day. Adding the time spent in one day making Chip One to the time spent making Chip Two, we have the linear inequality: 5c1 + 10c2 ≤ 1,440. This example is simple but not far-fetched. Linear inequalities that express limits or constraints on time, material, or other valuables appear constantly in the solution of real-world business and finance problems. Such problems are often solved using the technique called "linear programming." RECOVERING HUMAN MOTION FROM VIDEOThere is much interest today in teaching computers how to track human motion from video cameras. To track human motion successfully, a computer must be able to pick human forms out of all the other information in a moving image, like the video of a dance or a football game. It should then be able to describe what the human being has done in words, or be able to move a mathematical model or virtual puppet to re-create the motion it has observed. The ability to track and then describe or reproduce human motion mathematically is used in video games, virtual reality, formation analysis in sports, and in other ways. One method used to track motion is the identification of "feature points" on the subject—idealized dots or spots on the surface of the subject's body, say on their helmet or elbow or knee. These feature points are then tracked in the video images recorded by several video cameras. Since each video image is two-dimensional (flat), the location of each feature point in the image at any one time can be described by two numbers, an x coordinate that says how far from the left-hand edge of the image the point is and a y coordinate that says how high up it us from the bottom edge of the image. If there are, say, 40 feature points on the subject, there are then 80 numbers that describe where the feature points are at a particular moment of time in the video from a single camera, and 3 × 80 = 240 numbers to describe where the feature points in the video from 3 cameras. These numbers are put into a matrix. Using linear mathematics methods of matrix algebra, this matrix is separated into two matrices, an M matrix that describes how the cameras are pointing (which we don't really care about) and an S matrix that describes the true arrangement in space of the feature points. The S matrix records how the subject is positioned in space at that moment. A whole series of S matrices, one for each frame of video, describes how the subject's body moves through space over time. Motion capture and analysis using linear algebra is used in computer-animated movies such as Polar Express (2004), where live actors' motions were recorded by computers and then used to animate digital figures. VIRTUAL TENNISThe ongoing explosion in computer power makes possible the crafting of "virtual" worlds in which a game-player, scientist, or other user can experience the illusion of movement and exploration. In most virtual-world or virtual-reality systems, a headset replaces the scene that the user would otherwise see with computer-generated scenes. But the sense of touch is not so easy to fool. One approach, in virtual tennis, is to have a computer read position information from a racket grip held in the player's hand. A rod is also attached to the "racket." When the player sees a ball coming in the virtual world which the headset shows to them, they swing at the ball. The computer senses the forces that the player's hand exerts on the racket grip, as well as the position of the racket in three-dimensional space, and calculates whether they player is going to succeed in hitting the virtual ball. If they do, the computer sends a shock along a rod connected to the player's racket so that they can feel the impact of the ball—which does not physically exist—hitting the racket. Such systems are already becoming commercially available.
All the computations performed by the computer in such a game involve vectors and linear mathematics. The position of the racket in space is characterized by a set of three-dimensional vectors; the force of the player's grip, the velocity of the ball, and other variables are also represented by vectors. Furthermore, the computer must calculate what racket positions are "feasible" for the system, that is, what positions the rod and wiresattached to the racket can allow. This is done using matrix algebra. LINEAR PROGRAMMINGLinear inequalities (see sidebar) are important to the problem-solving method known as "linear programming." In a linear programming problem, linear equalities and linear inequalities are combined into a system (that is, they all involve the same variables or unknowns). The first step in a linear programming problem is to define a linear equation that describes something which we want to minimize (expenses, say) and as many linear inequalities as we need to describe the bounds on our resources: for instance, that there are only so many minutes in a day, or pounds of Ingredient Z available, or dollars in the bank available for investment. Each linear inequality is then converted into a linear equality. For example, the inequality 50x1 + 12x2 ≤ 100 really says that 50x1 + 12x2 is less than 100 by some unknown amount (maybe 0). This is the same as saying that an 50x1 + 12x2 plus an unknown quantity equals 100. If we name this third unknown M, we can turn the inequality into an equation: 50x1 + 12x2 + M = 100. When all linear inequalities have been turned into linear equations, we then use matrix algebra methods (which are described in many textbooks) to solve the system and find out the best way to run our business. Linear programming is used by real-life organizations, especially businesses and the military. An example is the use of processing stations in semiconductor manufacturing plants. These plants make the circuit-covered "chips" that run all complex electronic devices, including computers. Many thin layers of material have to be built up on each chip, and each layer requires many stages of optical and chemical processing. In fact, more resources are consumed in making the tiny chips in a desktop computer than in making all the rest of the computer put together. Manufacturers are therefore keen to use their chip-making factories efficiently. A processing station in a chip factory is a large, complex device that performs one step at a time in the chip-making process. Instead of having hundreds of stations, one for every step, it is cheaper to re-fit each station (change some of its parts) occasionally so that it can do a different step. But refitting a station takes time; it would be unprofitable to refit a station every single time it performed a step. How many batches of chips should a station process before being refitted for another step? Linear programming is used to answer this question, telling the manufacturer how to schedule steps and stations for maximum profit. LINEAR REPRODUCTION OF MUSICIf a musician plays two notes in a recording studio, one twice as loud as the other, you want two notes come out of your stereo's speakers so that the one is twice as loud as the other. If graphed on paper, this relationship between live performance and ideal playback is a straight line—a linear function. A great deal of mathematical design work goes into making sound-reproduction systems as linear as possible. Maximizing ProfitsThe technique known as "linear programming" combines linear equations with linear inequalities (see sidebar) to find the best way of using limited resources. It is used mostly by large organizations, such as corporations or the military, to minimize operating costs. Banks use linear programming to process checks more efficiently. In particular, they wanted to minimize "float." Float is the amount of money represented by uncancelled checks—checks that have been received by the bank but for which the money has not yet been collected. Float is detrimental to profit because it represents money in limbo; the bank cannot make money on that money (invest it) until the check has cleared. What should a bank do to minimize float without spending so much doing it that the cure is worse than the disease? When checks are received they are "encoded," that is, marked with magnetic ink by a machine. This is the first step in clearing the check. Banks realized they needed to encode checks as quickly as possible without hiring too many machines and clerks, so mathematicians and computer specialists set up a linear programming problem to model the situation. That is, they organized float, encoding machines costs, wages and hours for clerks, and other relevant variables as a set of linear equations and inequalities, and solved this system using linear algebra. The solution showed banks how many full- and part-time clerks to assign to how many shifts on how many machines in order to minimize float. Although there are increasingly high-tech ways to digitize information and handle checks, many financial institutions still use linear programming to save money and increase profits. But nonlinearity—electronic behavior that is not linear—has its uses, too. The rough sound of a rock guitar is produced by feeding an electrical signal derived from the guitar's strings into a circuit that does not respond linearly. That is, the original signal looks like a complicated wave or series of up-and-down wiggles; when two Where to Learn MoreBooksBudnick, Frank S. Finite Mathematics with Applications. New York: McGraw-Hill, 1985. Lay, David C. Linear Algebra and its Applications, 2nd ed. New York: Addison-Wesley, 1999. |
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