Linear equation

a₁x₁+a₂x₂+¼+a_nx_n = b

variables

x₁,¼,x_n

coefficients

a₁,¼,a_n

System of linear equations

(linear system)

a collection of linear equations in the same variables

solution set

set of all solutions of the system

equivalent systems

two systems of linear equations with the same solution set

consistent system

a linear system with at least one solution

inconsistent system

a linear system with no solution.

Matrix

a rectangular array (of numbers)

coefficient matrix

given a system of linear equations, a matrix whose entries are the coefficients of each variable aligned in columns.

augmented matrix

the matrix obtained by appending a column containing the constant terms of a system to the coefficient matrix

size

the number of rows (m) and columns (n) of a matrix

m×n matrix

a matrix with m rows and n columns

Elementary row operations

transformations of a matrix that do not change the solution set of the corresponding linear system

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

Echelon form

of a matrix

1. all nonzero rows are above any row of all zeros

2. each leading (nonzero) entry of a row is in a column to the right of the leading entry of the row above it

3. all entries in a column below a leading entry are zero

Reduced echelon form

of a matrix in echelon form

1. the leading entry in each nonzero row is 1

2. each leading 1 is the only nonzero entry in its column

Pivot

a nonzero number in a pivot position that is used to create zeroes via row operations

pivot position

a location (a row and a column) in a matrix that corresponds to a leading entry in an echelon form of the matrix

pivot column

a column in a matrix that contains a pivot position

Basic variables

variables corresponding to pivot columns in a matrix

Free variables

variables not corresponding to pivot columns

General solution

explicit description of all solutions of a linear system

Footnotes:

¹ Linear Algebra and its Applications, by David C. Lay, Sections 1.1 and 1.2.