Mathematics Foundations/6.3 Systems of Linear Equations

6.3 Systems of Linear Equations 6.3.1 Introduction to Linear Systems A system of linear equations consists of one or more linear equations involving the same set of variables. In general, a system of linear equations in unknowns can be written as: where are the coefficients, are the variables, and are the constants. 6.3.1.1 Matrix Representation A system of linear equations can be represented in matrix form as: where: is the coefficient matrix is the vector of variables is the vector of constants Explicitly: 6.3.1.2 Augmented Matrix The augmented matrix of a system is formed by appending the constant vector to the coefficient matrix : 6.3.2 Types of Linear Systems A system of linear equations can be classified based on the number of solutions: Consistent system: Has at least one solution * Uniquely determined: Has exactly one solution * Underdetermined: Has infinitely many solutions Inconsistent system: Has no solution The classification depends on the relationship between the coefficient matrix and the augmented matrix : If , the system has a unique solution If , the system has infinitely many solutions If , the system has no solution 6.3.3 Solving Linear Systems 6.3.3.1 Gaussian Elimination Gaussian elimination is a systematic method for solving systems of linear equations by transforming the augmented matrix into row echelon form through elementary row operations: 1. Arrange the equations so that the first equation has a non-zero coefficient for 2. Eliminate from all equations below the first equation 3. Arrange the remaining equations so that the second equation has a non-zero coefficient for 4. Eliminate from all equations below the second equation 5. Continue this process until the augmented matrix is in row echelon form 6. Use back-substitution to find the values of the variables Example: Solve the system Step 1: Write the augmented matrix Step 2: Apply row operations to get row echelon form Step 3: Back-substitution Solution: 6.3.3.2 Gauss-Jordan Elimination Gauss-Jordan elimination extends Gaussian elimination by continuing the process to transform the augmented matrix into reduced row echelon form: 1. Perform Gaussian elimination to get row echelon form 2. Starting from the bottom row, eliminate all non-zero entries above each pivot 3. Scale each row to make the pivot equal to 1 A matrix is in reduced row echelon form when: It is in row echelon form Every pivot is 1 Each pivot is the only non-zero entry in its column Example: Using the same system as above, continue from the row echelon form: Step 1: Scale the third row Step 2: Eliminate from the first and second rows Step 3: Scale the second row Step 4: Eliminate from the first row Solution: 6.3.3.3 Matrix Inverse Method For a system where is a square invertible matrix, the solution can be found by: Example: Solve the system Step 1: Identify and Step 2: Find (using methods from Section 6.2) Step 3: Calculate Solution: 6.3.3.4 Cramer's Rule For a system where is a square invertible matrix, Cramer's rule states that each variable can be found by: where is the matrix obtained by replacing the -th column of with the vector . Example: Using the same system as above: Step 1: Calculate Step 2: Calculate Step 3: Calculate Step 4: Calculate Step 5: Calculate Solution: 6.3.4 Homogeneous Systems A homogeneous system of linear equations has the form , where is the zero vector. Properties of homogeneous systems: A homogeneous system always has the trivial solution A homogeneous system has non-trivial solutions if and only if (for square matrices) or (for matrices) The set of all solutions to a homogeneous system forms a subspace of , specifically the nullspace of 6.3.4.1 Finding the General Solution To find the general solution of a homogeneous system: 1. Perform Gaussian elimination to get row echelon form 2. Identify the free variables (variables that are not leading variables in the row echelon form) 3. Express the leading variables in terms of the free variables 4. Write the general solution as a linear combination of vectors, where each vector corresponds to setting one free variable to 1 and the others to 0 Example: Find the general solution of the homogeneous system Step 1: Write the augmented matrix Step 2: Apply row operations to get row echelon form Step 3: Further simplify Step 4: Identify the leading variables ( and ) and free variables ( and ) Step 5: Express the leading variables in terms of the free variables Step 6: Write the general solution The general solution is , where and . 6.3.5 Applications of Linear Systems 6.3.5.1 Network Analysis In electrical circuit analysis, Kirchhoff's laws lead to systems of linear equations where the variables represent currents or voltages. 6.3.5.2 Leontief Input-Output Model In economics, the Leontief model describes the interdependence of industries in an economy, leading to systems of linear equations. 6.3.5.3 Linear Regression In statistics, linear regression involves solving a system of linear equations to find the best-fit line or hyperplane for a set of data points. 6.3.5.4 Computer Graphics In computer graphics, transformations such as rotation, scaling, and translation are represented by systems of linear equations. 6.3.6 Computational Considerations 6.3.6.1 Condition Number The condition number of a matrix , denoted by , measures the sensitivity of the solution of to changes in : A large condition number indicates an ill-conditioned system, which may lead to numerical instability. 6.3.6.2 Iterative Methods For large sparse systems, iterative methods such as Jacobi, Gauss-Seidel, and Conjugate Gradient are often more efficient than direct methods. 6.3.7 Exercises 1. Solve the following system of linear equations using Gaussian elimination: 2. Determine the values of for which the following system has (a) no solution, (b) a unique solution, (c) infinitely many solutions: 3. Find the general solution of the homogeneous system: 4. Use Cramer's rule to solve: 5. A factory produces three types of products: A, B, and C. Each product requires certain amounts of raw materials: 1, 2, and 3 units of material X; and 2, 1, and 1 units of material Y, respectively. If the factory has 140 units of material X and 90 units of material Y available, and wants to produce twice as many units of product A as product B, how many units of each product should be produced to use all available materials?