key term - Linear System

Definition

A linear system is a collection of linear equations that describe a relationship between multiple variables. It is a fundamental concept in linear algebra and is widely used in various fields, including mathematics, physics, engineering, and economics.

5 Must Know Facts For Your Next Test

1. A linear system can be represented in matrix form, where the coefficients of the variables and the constants are arranged in a matrix.
2. The solution to a linear system can be found using various methods, such as Gaussian elimination, matrix inverse, or Cramer's rule.
3. The number of linearly independent equations in a linear system determines the number of variables that can be uniquely determined.
4. If a linear system has a unique solution, it is said to be consistent; if it has no solution, it is said to be inconsistent; and if it has infinitely many solutions, it is said to be dependent.
5. The rank of the coefficient matrix of a linear system is an important concept that determines the number of linearly independent equations and the number of variables that can be uniquely determined.

Review Questions

• Explain how a linear system can be represented in matrix form and how this representation is used to solve the system.
  • A linear system can be represented in matrix form by arranging the coefficients of the variables and the constants into a matrix. The coefficients of the variables form the coefficient matrix, and the constants form the right-hand side of the system. This matrix representation allows for the use of matrix operations, such as Gaussian elimination, to solve the system. By performing row operations on the augmented matrix, the system can be transformed into an equivalent system that is easier to solve, ultimately leading to the determination of the values of the variables that satisfy the original linear system.
• Describe the different types of solutions that a linear system can have and the conditions that determine each type of solution.
  • A linear system can have three types of solutions: unique solution, no solution, or infinitely many solutions. If the system is consistent, meaning the equations are linearly independent, then the system will have a unique solution. If the system is inconsistent, meaning the equations are contradictory, then the system will have no solution. If the system is dependent, meaning the equations are linearly dependent, then the system will have infinitely many solutions. The number of linearly independent equations in the system and the rank of the coefficient matrix are key factors in determining the type of solution the system will have.
• Analyze the relationship between the rank of the coefficient matrix of a linear system and the number of variables that can be uniquely determined in the system.
  • The rank of the coefficient matrix of a linear system is a crucial concept that determines the number of variables that can be uniquely determined in the system. The rank of a matrix represents the number of linearly independent rows or columns in the matrix. If the rank of the coefficient matrix is equal to the number of variables in the system, then all the variables can be uniquely determined, and the system has a unique solution. However, if the rank of the coefficient matrix is less than the number of variables, then the system will have infinitely many solutions, and only a subset of the variables can be uniquely determined. Understanding the relationship between the rank of the coefficient matrix and the number of variables that can be uniquely determined is essential for solving linear systems effectively.

Related terms

Linear Equation:

A linear equation is an equation that can be written in the form $ax + by + c = 0$, where $a$, $b$, and $c$ are constants and $x$ and $y$ are variables.

Matrix:

A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns, that can be used to represent and manipulate linear systems.

Augmented Matrix:

An augmented matrix is a matrix formed by appending the coefficients of the constants on the right-hand side of a system of linear equations to the matrix of coefficients of the variables.