Linear - Ch. 1.1 (Systems of Linear Equations)

linear equation

system of linear equations (or linear system)

collection of one or more linear equations involving the same variables

solution of a linear system

a list of s numbers that makes each equation a true statement when s values are substituted for x values

solution set

set of all possible solutions of a linear system

two linear systems are called ____ if they have the same solution set

equivalent

types of solutions for linear equations

no solution or

exactly one solution or

infinitely many solutions

a system of linear equations is said to be ______ if it has one solution or infinitely many solutions

consistent

a system is ______ if it has no solution

inconsistent

matrix

the essential information of a linear system can be recorded compactly in a rectangular array

the matrix of coefficients

taking the coefficients of the system with the coefficients of each variable aligned in columns

augmented matrix

matrix of coefficients + column containing the constants from right sides of equations

basic strategy to solving a linear system

replace one system with an equivalent system (one with the same solution set) that is easier to solve

elementary row operations

replacement

interchange

scaling

replacement

replace one row by the sum of itself and a multiple of another row

interchange

interchange two rows

scaling

multiply all entries in a row by a nonzero constant

two matrices are called _______ if there is a sequence of elementary row operations that transforms one matrix into the other

row equivalent

if the augmented matrices of two linear systems are row equivalent, then the two systems have the same _______

solution set