Solving Systems of Linear Equations | Unraveling the Knots

The field of linear algebra is a fundamental tool in many areas of mathematics and science. One of the most important concepts in linear algebra is the solution of systems of linear equations. These systems can be represented by matrices and vectors, and the goal is to find the values of the variables that satisfy all of the equations in the system. In this document, we will delve into the various methods and techniques used to solve systems of linear equations, as well as the underlying theory and properties that make these methods possible.

 

Introduction

The study of systems of linear equations is an essential topic in linear algebra. These systems can be found in various fields, such as physics, engineering, and economics. The solution of such systems enables us to understand and analyze the relationships between different variables and make predictions based on that knowledge. In this document, we will explore the different techniques used to solve systems of linear equations, including Gaussian elimination, Gauss-Jordan elimination, and matrix inversion. We will also examine the properties and theorems that make these methods possible, as well as the applications of these techniques in different fields.

 

Definitions

System of Linear Equations

A system of linear equations is a collection of one or more linear equations involving the same variables. A linear equation is an equation of the form $ax + b = c$, where $a$, $b$, and $c$ are constants. A system of linear equations can be represented in matrix form as $Ax = b$, where $A$ is a matrix of coefficients, $x$ is a vector of variables, and $b$ is a vector of constants.

Gaussian Elimination

Gaussian elimination is a method of solving systems of linear equations by transforming the matrix of coefficients into an upper-triangular form. This is done by applying a sequence of row operations, such as swapping rows, multiplying rows by a constant, or adding a multiple of one row to another. Once the matrix is in upper-triangular form, the solution can be found by back-substitution.

Gauss-Jordan Elimination

Gauss-Jordan elimination is a variation of Gaussian elimination that is used to find the inverse of a matrix. The method is the same as Gaussian elimination, but the goal is to transform the matrix into an identity matrix, rather than an upper-triangular matrix. Once the matrix is in identity form, the inverse can be read off the diagonal elements.

Matrix Inversion

Matrix inversion is the process of finding the inverse of a matrix. A matrix is said to be invertible if there exists a matrix such that $A^{-1}A = AA^{-1} = I$, where $I$ is the identity matrix. The inverse of a matrix can be used to solve systems of linear equations, as well as to perform other operations in linear algebra.

 

Theorems

The Invertibility Theorem

The Invertibility Theorem states that a matrix is invertible if and only if its determinant is nonzero. The determinant is a scalar value that can be calculated from the matrix of coefficients. If the determinant is zero, then the matrix is singular and does not have an inverse.

The Rank Theorem

The Rank Theorem states that the rank of a matrix is equal to the number of non-zero rows in the row echelon form of the matrix. The rank of a matrix is a measure of the number of linearly independent rows or columns in the matrix. If the rank of a matrix is less than the number of variables, then the system of equations is underdetermined and has infinitely many solutions. If the rank of a matrix is equal to the number of variables, then the system of equations is determined and has a unique solution. If the rank of a matrix is greater than the number of variables, then the system of equations is overdetermined and has no solutions.

 

Properties

Linearity

Linearity is a property of systems of linear equations that states that the sum of any two solutions is also a solution, and that any solution can be multiplied by a scalar to obtain another solution. This property is a consequence of the fact that the matrix of coefficients and the vector of constants are linear functions of the variables.

Homogeniety

Homogeneity is a property of systems of linear equations that states that if a solution exists, then the system has infinitely many solutions, which can be obtained by adding any non-zero multiple of the original solution to itself. This property is a consequence of the fact that the matrix of coefficients and the vector of constants are homogeneous functions of the variables.

 

Example

Consider the following system of linear equations:

$3x + 2y - z = 1$

$2x - 2y + 4z = 2$

$-x + \frac{1}{2}y - z = 0$

We can represent this system in matrix form as $Ax = b$, where

$A = \begin{bmatrix} 3 & 2 & -1 \\ 2 & -2 & 4 \\ -1 & \frac{1}{2} & -1 \end{bmatrix}$, $x = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$, $b = \begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix}$

To solve this system, we can use Gaussian elimination to reduce the matrix $A$ to an upper-triangular form and then use back-substitution to find the values of $x$, $y$, and $z$. The steps for this method are as follows:

 

Multiply row 1 by $\frac{1}{3}$ to get $x$ alone.

$\begin{bmatrix} 1 & \frac{2}{3} & -\frac{1}{3} \\ 2 & -2 & 4 \\ -1 & \frac{1}{2} & -1 \end{bmatrix}$

 

Subtract 2 times row 1 from row 2 to eliminate the $x$ term.

$\begin{bmatrix} 1 & \frac{2}{3} & -\frac{1}{3} \\ 0 & -\frac{8}{3} & \frac{10}{3} \\ -1 & \frac{1}{2} & -1 \end{bmatrix}$

 

Add row 1 to row 3 to eliminate the $x$ term.

$\begin{bmatrix} 1 & \frac{2}{3} & -\frac{1}{3} \\ 0 & -\frac{8}{3} & \frac{10}{3} \\ 0 & \frac{5}{6} & \frac{2}{3} \end{bmatrix}$

 

Subtract $\frac{5}{6}$ times row 3 from row 2 to eliminate the $z$ term.

$\begin{bmatrix} 1 & \frac{2}{3} & -\frac{1}{3} \\ 0 & 0 & \frac{20}{9} \\ 0 & \frac{5}{6} & \frac{2}{3} \end{bmatrix}$

 

The matrix is now in upper-triangular form, and we can use back-substitution to find the values of $x$, $y$, and $z$.

$z = \frac{20}{9}$

$y = 0$

$x = -\frac{1}{3}$

Therefore, the solution of the system of linear equations is $x = -\frac{1}{3}$, $y = 0$, $z = \frac{20}{9}$.

 

Applications

Solving systems of linear equations has many practical applications in various fields such as physics, engineering, economics, and computer science. Some examples of applications include:

• In physics, solving systems of linear equations can be used to model the motion of particles in a system, such as the motion of planets in a solar system.
• In engineering, solving systems of linear equations can be used to design and analyze structures, such as bridges and buildings.
• In economics, solving systems of linear equations can be used to model and analyze economic systems, such as supply and demand.
• In computer science, solving systems of linear equations can be used to solve optimization problems, such as finding the shortest path in a graph.

 

Conclusion

In this document, we have discussed the fundamental concepts of solving systems of linear equations. We have defined the basic terms, discussed theorems and properties related to the subject, provided a specific example and its solution, and highlighted some of the applications of solving systems of linear equations. The method of solving a system of equations is a fundamental tool for solving mathematical and real-world problems, and it is an essential topic for students of mathematics, physics, engineering, economics, and computer science.