linear system

(noun)

A mathematical model of a system based on the use of a linear operator.

Examples of linear system in the following topics:

  • Inconsistent and Dependent Systems

    • Two properties of a linear system are consistency (are there solutions?
    • In mathematics, a system of linear equations (or linear system) is a collection of linear equations involving the same set of variables.
    • A linear system may behave in any one of three possible ways:
    • This is an example of equivalence in a system of linear equations.
    • A linear system is consistent if it has a solution, and inconsistent otherwise.

  • Introduction to Systems of Equations

    • A system of linear equations consists of two or more linear equations made up of two or more variables such that all equations in the system are considered simultaneously.
    • Some linear systems may not have a solution and others may have an infinite number of solutions.
    • For example, consider the following system of linear equations in two variables.
    • In this example, the ordered pair (4, 7) is the solution to the system of linear equations.
    • In general, a linear system may behave in any one of three possible ways:

  • Nonlinear Systems of Equations and Problem-Solving

    • As with linear systems, a nonlinear system of equations (and conics) can be solved graphically and algebraically for all its variables.
    • In a system of equations, two or more relationships are stated among variables.
    • As with linear systems of equations, substitution can be used to solve nonlinear systems for one variable and then the other.
    • Solving nonlinear systems of equations algebraically is similar to doing the same for linear systems of equations.
    • We can solve this system algebraically by using equation (1) as a substution.

  • Inconsistent and Dependent Systems in Two Variables

    • For linear equations in two variables, inconsistent systems have no solution, while dependent systems have infinitely many solutions.
    • Recall that linear system may behave in any one of three possible ways:
    • Also recall that each of these possibilities corresponds to a type of system of linear equations in two variables.
    • We will now focus on identifying dependent and inconsistent systems of linear equations.
    • A linear system is consistent if it has a solution, and inconsistent otherwise.

  • Matrices and Row Operations

    • The row space of a matrix is the set of all possible linear combinations of its row vectors.
    • If the rows of the matrix represent a system of linear equations, then the row space consists of all linear equations that can be deduced algebraically from those in the system.  
    • Since the matrix is essentially the coefficients and constants of a linear system, the three row operations preserve the matrix.  
    • Also, when solving a system of linear equations by the elimination method, row multiplication would be the same as multiplying the whole equation by a number to obtain additive inverses so that a variable cancels.  
    • Because these operations are reversible, the augmented matrix produced always represents a linear system that is equivalent to the original.

  • Nonlinear Systems of Inequalities

    • Systems of nonlinear inequalities can be solved by graphing boundary lines.
    • A system of inequalities consists of two or more inequalities, which are statements that one quantity is greater than or less than another.
    • All points below the line $y=x+2$ satisfy the linear equality, and all points above the parabola $y=x^2$ satisfy the parabolic nonlinear inequality.
    • This area is the solution to the system.
    • Whereas a solution for a linear system of equations will contain an infinite, unbounded area (lines can only pass one another a maximum of once), in many instances, a solution for a nonlinear system of equations will consist of a finite, bounded area.

  • Matrix Equations

    • Matrices can be used to compactly write and work with systems of multiple linear equations.
    • This is very helpful when we start to work with systems of equations.
    • Solving a system of linear equations using the inverse of a matrix requires the definition of two new matrices: $X$ is the matrix representing the variables of the system, and $B$ is the matrix representing the constants.
    • To solve a system of linear equations using an inverse matrix, let $A$ be the coefficient matrix, let $X$ be the variable matrix, and let $B$ be the constant matrix.
    • Thus, we want to solve a system $AX=B$, for $X$.  

  • Application of Systems of Inequalities: Linear Programming

    • Linear programming involves finding an optimal solution for a linear equation, given a number of constraints.
    • A common application of systems of inequalities is linear programming.
    • Linear programming is a mathematical method for determining a way to achieve the best outcome for some list of requirements represented as linear relationships.
    • An example where linear programming would be helpful to optimize a system of inequalities is as follows:
    • Use the Simplex Method to solve applications of systems of linear inequalities

  • Graphs of Linear Inequalities

    • Graphing linear inequalities involves graphing the original line, and then shading in the area connected to the inequality.
    • Introduction to Graphs of Linear Inequalities:  Single Inequality in Two Variables
    • These overlaps of the shaded regions indicate all solutions (ordered pairs) to the system.
    • The brown overlapped shaded area is the final solution to the system of linear inequalities because it is comprised of all possible solutions to $y<-\frac{1}{2}x+1$ (the dotted red line and red area below the line) and $y\geq x-2$ (the solid green line and the green area above the line).  
    • The origin is a solution to the system, but the point $(3,0)$ is not.

  • Linear and Quadratic Equations

    • In a set simultaneous equations, or system of equations, multiple equations are given with multiple unknowns.
    • A solution to the system is an assignment of values to all the unknowns so that all of the equations are true.
    • Two kinds of equations are linear and quadratic.
    • Linear equations can have one or more variables.
    • Linear equations do not include exponents.