Applied Finite Mathematics

The section begins by establishing fundamental matrix terminology. It defines key concepts such as matrix entries (or elements), matrix dimensions (number of rows and columns), square matrices (equal rows and columns), zero matrices (all entries are zero), and identity matrices (ones along the main diagonal, zeros elsewhere). The significance of matrix dimensions is highlighted, emphasizing that a 3x4 matrix is distinct from a 4x3 matrix. Row matrices (or row vectors) and column matrices (or column vectors) are also defined, along with the condition for matrix equality (same size and corresponding entries). This foundational knowledge is essential for understanding subsequent matrix operations and the Gauss-Jordan method.

The Gauss-Jordan method is introduced as a technique for solving systems of linear equations. The process starts by expressing the system of equations as an augmented matrix. This matrix then undergoes a series of row operations – elementary transformations applied to the rows – to systematically reduce it to an equivalent, simpler form. These operations maintain the solution to the system throughout the reduction. The ultimate goal is to arrive at a matrix where the solution becomes evident by inspection. The text emphasizes that the solution remains consistent throughout the row operations, highlighting the equivalence between the original system and the simplified, reduced form.

The concept of reduced row echelon form is explained as the target form for the Gauss-Jordan method. A matrix is in reduced row echelon form when the first non-zero entry in each row is 1 (a leading 1), all other entries in the column containing a leading 1 are zero, and the leading 1 in each row is to the right of the leading 1 in the row above it. Rows containing only zeros are at the bottom. The text provides an example where the final reduced row echelon form clearly shows the solution: x = 1, y = 2, and z = 3. The process uses three basic row operations: swapping two rows, multiplying a row by a non-zero scalar, and adding a multiple of one row to another. The method systematically applies these transformations to achieve the reduced row echelon form.

This subsection defines key terms associated with the Gauss-Jordan elimination process. Pivoting is described as the process of obtaining a 1 in a specific location (the pivot element) within the matrix and then making all other entries in that column zero. The pivot element is the number that's transformed into a 1, and its row is the pivot row. The target row is the row to which a multiple of the pivot row is added to obtain a zero in a particular entry. These terms are crucial for understanding the steps and strategies involved in performing the row operations effectively, ensuring that the matrix is efficiently transformed into reduced row echelon form to find the solution to the system of linear equations. The text explains how these operations are used to systematically eliminate variables and solve for the unknowns.

The section addresses scenarios where a system of linear equations has infinitely many solutions. In these cases, the solution is often expressed in parametric form. This involves assigning an arbitrary constant (often denoted as 't') to one of the variables and then solving for the remaining variables in terms of this constant. The text gives an example where letting y = t leads to a solution expressed as x = 7 - t, indicating that all ordered pairs of the form (7 - t, t) satisfy the given system. This demonstrates how to handle and represent solutions when a unique solution doesn't exist, highlighting a key difference from systems with unique solutions directly obtained from the reduced row echelon form of the augmented matrix.

This section describes a method for using matrices in cryptography to encode and decode secret messages. The process begins by converting the secret message into a numerical string. Each letter in the message is assigned a unique number. This numerical representation is then transformed into a new set of numbers by multiplying the original numerical string by a chosen square matrix. This square matrix must have an inverse for decoding purposes. The resulting set of numbers represents the encoded message, suitable for secure transmission. Decoding the message involves multiplying the coded numerical string by the inverse of the initially selected square matrix, to retrieve the original numerical string. Finally, by reversing the initial letter-to-number assignment, the original message is recovered. The method's efficacy relies on the secure transmission of the encoding matrix and the existence of its inverse.

The core of the cryptographic method hinges on the use of square matrices and their inverses. A square matrix is a matrix with an equal number of rows and columns. The selection of the square matrix is arbitrary but crucial for the effectiveness of the encoding. The existence of an inverse matrix is absolutely necessary. The inverse matrix is used to decode the encrypted message; multiplying the encoded numerical string by the inverse yields the original numerical string, allowing for the recovery of the original message. The choice of a suitable square matrix that possesses an inverse is, therefore, a vital step in this encryption-decryption scheme. The security of the method relies on keeping the specific square matrix and its inverse confidential.

This section introduces Wassily Leontief's input-output models, a significant contribution to economics for which he received the Nobel Prize in 1973. These models utilize matrices to represent the interconnectedness of an economy's various sectors. Each sector produces goods and services not only for its own use but also for other sectors, creating a system of interdependence. The fundamental principle is that the total input to the system always equals the total output. The models are designed to analyze the flow of goods and services between these sectors, providing insights into economic dependencies and overall system behavior. The text mentions both closed and open models, indicating variations in how the models handle external factors influencing economic interactions and production.

To illustrate the application of Leontief's input-output models, a practical example is presented involving Chris, a carpenter, and Ed, an electrician. They mutually help each other with repairs on their respective houses. Chris spends 60% of his time on his own house and 40% on Ed's. Ed, conversely, divides his time equally between his own house and Chris's. This scenario highlights the interdependency between the two individuals as economic actors. This example serves as a simplified model of economic exchange. The problem focuses on determining the fair compensation for each individual, given their contribution to the work and the original agreement of approximately $1000 each. This illustrates how a Leontief-style model, even on a very small scale, can account for interdependence and calculate fair distribution of value.

The concept is further illustrated with additional examples involving students helping each other with studies and tutoring. These examples build upon the previous house repair example to show additional applications of input-output analysis. Chris, Bob, and Matt each dedicate portions of their study time to their own subjects and helping others. For instance, Chris spends 30% of his time on chemistry, 15% helping Bob, and 25% helping Matt. Similarly, Bob dedicates time to biology and assisting Chris and Matt. This exemplifies an extension of input-output model concepts to a different setting, such as time spent on tasks rather than goods and services. These types of scenarios effectively demonstrate the versatility and applicability of input-output models to analyze various interconnected situations involving resource allocation, time management, and the value of mutual assistance among multiple parties.

This section introduces the core concepts of linear programming. A linear programming problem involves finding the extreme value (either maximum or minimum) of a linear function, known as the objective function, while satisfying certain constraints. These constraints are conditions that must be met and are often expressed as inequalities. The problems are categorized as either maximization or minimization problems, both falling under the umbrella of optimization problems. The text explains that for problems with only two variables, a graphical method can be employed to find the solution. This lays the groundwork for understanding how to formulate and approach these types of problems, establishing the key elements that define a linear programming problem: the objective function and the constraints.

The section demonstrates how to solve linear programming maximization problems using a graphical approach. The method involves plotting the constraints as inequalities on a graph, identifying the feasible region (the area where all constraints are satisfied), and then finding the point within the feasible region that maximizes the objective function. The extreme value of the objective function always occurs at a vertex (corner point) of the feasible region. An example is provided involving a factory manufacturing regular and premium gadgets. Each gadget requires assembly and finishing time, subject to constraints on total available hours for each operation and an upper limit on the total number of gadgets produced per day. The objective is to maximize profit given a profit margin for each type of gadget. This illustrates the application of graphical linear programming to a real-world production scenario.

Minimization linear programming problems are addressed, highlighting their similarity to maximization problems in terms of solution methodology. The key difference lies in the form of the constraints: minimization problems typically involve constraints of the form ax + by ≥ c, as opposed to ax + by ≤ c for maximization problems. Consequently, the feasible region for minimization problems extends infinitely to the upper right of the first quadrant. However, the solution strategy remains the same: the objective function is minimized at a vertex of the feasible region closest to the origin. The text illustrates this with the example of Professor Symons hiring John and Mary to grade homework papers. The objective is to minimize grading costs, considering each student's hourly rate and grading speed, while meeting a minimum number of papers to be graded and a minimum number of hours each student must work. This applies the graphical method to a resource allocation problem focusing on cost minimization.

Several examples illustrate diverse applications of linear programming, emphasizing its use in various fields. These examples highlight both maximization and minimization problems. One example involves Mr. Shoemacher's investment strategy, aiming to maximize annual yield from two mutual funds with investment amount constraints. Another example focuses on a computer store's goal to minimize sales commissions while meeting sales targets, considering constraints on the ratio of desktops to laptops sold. Other scenarios include maximizing profits for a printer company considering production constraints on assembly, testing, and finishing time and minimizing the risk for Mr. Boutros when investing in two stocks. The examples also illustrate minimizing costs for a typing service using two part-time typists, maximizing profit for a department store selling two types of televisions, minimizing breakfast costs for John based on vitamin, mineral, and calorie requirements from two cereals, and minimizing grading time for a professor when assigning objective and recall quizzes. These diverse examples demonstrate the widespread applicability of linear programming in business, finance, and educational settings.

The section begins by explaining why the graphical method for solving linear programming problems is insufficient for problems with more than two variables. Real-world linear programming problems often involve thousands of variables, making algebraic methods computationally expensive and impractical. The text highlights the limitations of evaluating the objective function at every corner point of a high-dimensional feasible region. This leads to the introduction of the simplex method as a more efficient algorithmic approach designed for computer implementation. The simplex method offers a systematic way to find the optimal solution without needing to examine every corner point, making it far more suitable for large-scale problems than traditional methods.

A brief history of the simplex method is presented. It was developed by George Dantzig during World War II, aiding the Allied forces with transportation and scheduling. The text notes that while the ellipsoid algorithm (developed by Leonid Khachian) and Karmarkar's algorithm (developed by Narendra Karmarkar) were proposed as alternatives, the simplex method remains the most effective for many problems. Karmarkar's algorithm showed potential speed advantages in certain cases, but the simplex method continues to be widely used due to its overall efficiency and robustness across various linear programming problem types. This historical context adds weight to the method's importance and continued relevance in the field of optimization.

The core of the simplex method is described as an iterative process. Unlike methods that evaluate the objective function at every point in the feasible region, the simplex method starts at a corner point where all main variables are zero. It then systematically moves from one corner point to another, each move improving the value of the objective function. This iterative procedure continues until the optimal solution is reached. The method is highly efficient because it avoids unnecessary calculations, focusing only on improving solutions at each step. This highlights the significant computational advantage of the simplex method over brute-force approaches for solving large-scale linear programming problems.

The final part addresses two key aspects of the simplex method: the role of quotients and the selection of the pivot element. The text explains that finding the most negative entry in the bottom row helps to maximize the objective function. However, constraints limit the increase of the variable, and computing quotients helps determine the maximum allowable increase for the variables without violating these constraints. The smallest quotient identifies the row that will become the pivot row in the next iteration. Selecting the pivot element, which becomes a 1, enables the method to move to the next corner point and further enhance the objective function value. This systematically changes the number of units of variables, adding to one while reducing another, always progressing toward the optimal solution. The text explains that pivoting is the mechanism allowing for this iterative improvement and movement across corner points of the feasible region.

The section begins by defining interest as the cost of borrowing money, with the principal (or present value) representing the amount borrowed or loaned. Simple interest is calculated only on the original principal, while compound interest is calculated on the principal plus accumulated interest. The text emphasizes the difference between simple and compound interest using an example where an initial deposit of $200 accumulates to $233.28 in two years with compound interest at 8%, while it would only reach $232 with simple interest. This highlights the effect of earning interest on interest in compound interest calculations. The concept of effective interest rate is also introduced as a way to compare different interest schemes. The text briefly mentions that interest can be compounded at various intervals (yearly, semiannually, quarterly, monthly, daily, or even continuously).

The concept of an annuity is introduced as a sequence of fixed payments made at equal time intervals. The section focuses primarily on ordinary annuities. The text explains that annuities are different from lump-sum deposits, where a single deposit is made. In contrast, annuities involve numerous deposits made at regular intervals. The difference is crucial when calculating the future value or present value of an investment. The section aims to provide understanding of the fundamental characteristics of an annuity without delving into intricate formula derivations. The goal is for students to grasp the concepts rather than simply memorizing formulas, and to differentiate them from lump-sum investments.

The section uses an example to show how to determine the amount of monthly payments needed to match the future value of a lump-sum cash payment. Two scenarios are compared: Mr. Cash pays $15,000 cash for a car, while Mr. Credit makes monthly payments over five years. The objective is to determine the size of Mr. Credit's monthly payments (x) at a 9% interest rate, ensuring his future value equals Mr. Cash's $15,000 investment after five years. This exemplifies how to equate the future values of lump-sum and annuity-based investments. The text also discusses finding the present value of an annuity by considering the reverse problem: if monthly payments for a car are known, how much does the car cost? This connects the future value and present value concepts, showing how they are used to compare different payment scenarios and determine equivalent amounts across time.

This subsection focuses on calculating the remaining balance of a loan at a specific time, relevant when loans are paid off prematurely. The example involves Mr. Jackson, who has a loan with monthly payments of $1260 for 10 more years. The text shows how to determine the present value of the remaining payments to find the balance owed. This utilizes the present value of an annuity formula. Another application explores bond valuation. A bond is defined as a certificate of promise; a business typically sells bonds at a face value (e.g., $1000) for a certain period (e.g., 10 years). The bondholder receives periodic interest payments (e.g., $30 every six months). The text explains that bond prices fluctuate with changing market interest rates; the fair market value of the bond needs to be calculated based on current interest rates. These examples demonstrate the application of present and future value calculations in practical financial situations, such as loan repayments and bond pricing.

The section concludes by providing guidance on identifying the type of financial problem—lump-sum or annuity—presented in a problem statement. It highlights that annuity problems often contain words such as ‘each,’ ‘every,’ or ‘per,’ indicating recurring payments. Lump-sum problems, in contrast, involve a single deposit. This emphasizes the importance of carefully reading the problem statement to correctly determine whether a lump sum or an annuity is involved in the financial calculation. The ability to accurately classify the type of problem significantly affects which formula should be used to calculate the present value, future value, or other relevant financial metrics.

This section introduces Venn diagrams as a visual tool for representing relationships between sets. Developed by John Venn in the late 1800s, these diagrams use circles to represent sets, often enclosed within a rectangle representing the universal set. The intersection and union of sets are easily visualized using this method. The section emphasizes the use of Venn diagrams for sorting populations and counting objects. An example is provided where a survey of 38 people regarding car transmissions (automatic vs. standard) is analyzed using a Venn diagram. The diagram is used to organize the responses, showing the number of people who drive only automatic, only standard, or both types of transmissions. This showcases Venn diagrams as a useful tool for organizing and analyzing categorical data.

The use of tree diagrams for visualizing and counting possibilities in multi-step processes is explained. A tree diagram is presented as a method to represent the different pathways or choices in a sequential process. Each branch represents a choice, and the total number of possible outcomes is determined by multiplying the number of choices at each step. An example is provided involving a woman choosing an outfit (blouse, skirt, and pumps) with multiple choices for each item. This example demonstrates the effectiveness of tree diagrams in visualizing the number of possibilities when making consecutive choices. The total number of possible outfits (12) is obtained by multiplying the number of choices at each step (2 blouses * 3 skirts * 2 pumps).

The section implicitly touches upon principles of permutations and combinations through the examples of tree diagrams and seating arrangements. The tree diagram example illustrates how to count the number of possible arrangements, representing permutations of choices. A further example involving the arrangement of three building blocks (A, B, C) into sequences illustrates permutations where the order matters. The section then contrasts this with the problem of seating three people in a circle (circular permutations). It explains that the number of ways to arrange the people is smaller compared to a linear arrangement because rotating the arrangement in a circle does not produce a new, distinct arrangement. This shows the contrast between linear and circular permutations, emphasizing that the first person in a circular arrangement can be considered a placeholder, as their position doesn't create a distinct arrangement.