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Section 1.2 Terminology and notation

linear equation

an equation of the form

\begin{equation*} \text{number times variable plus number times variable plus }\dots \text{ equals number} \end{equation*}
system of linear equations

a (finite) collection of linear equations

Note 1.2.1.

We will often just say system of equations to mean a system of linear equations.

solution

a set of values that, when substituted in for the variables of a system of equations, satisfy all of the equations in the system simultaneously

solution set

the collection of all possible solutions to a system of equations

consistent system

a system of equations with at least one solution

inconsistent system

a system of equations with no solution

parametric equations

a collection of formulas, based on one or more parameters, for the variables in a linear system that represent all possible solutions to the system (as the parameters vary over all real numbers)

matrix

a rectangular array of numbers

augmented matrix

a matrix of the coefficients and constants in a linear system

matrix entry

one of the numbers in a matrix; sometimes also referred to as a matrix coefficient

row

a horizontal line of entries in a matrix

column

a vertical line of entries in a matrix

elementary row operations

the three basic operations that can be applied to an augmented matrix without changing the set of solutions to the corresponding linear system:

  1. swap the positions of two rows,
  2. multiply a row by a nonzero constant, and
  3. add a multiple of one row to another.