Section 2.2 Terminology and notation
- row echelon form
-
a matrix that has the following properties:
- if a row has nonzero entries, its first nonzero entry is a one (called a leading one),
- each leading one occurs in a column that is to the right of the column containing the leading one in the row above it, and
- zero rows appear below all nonzero rows
- reduced row echelon form
-
a row echelon form matrix that also has the following property:
- the leading one of each row has all other entries in the column that contains it equal to zero
Note 2.2.1.
The acronyms REF and RREF are commonly used for row echelon form and reduced row echelon form, respectively.
- row reduction
the process of using elementary row operations to reduce a matrix to REF or RREF
- row equivalent matrices
matrices where it is possible to obtain one from the other through a sequence of elementary row operations
- rank
the number of leading ones in the RREF of the matrix
- leading variables
the variables in a linear system whose columns in the RREF of the augmented matrix contain the leading one of some row
- free variables
the variables in a linear system that are not leading variables
- general solution
a set of parametric equations from which all solutions to a linear system can be obtain by choosing arbitrary values for the parameters
- homogeneous system
a linear system in which the “equals” column is all zeros
- coefficient matrix
the matrix for a linear system but without the “equals” column
- trivial solution
the obvious solution to a homogeneous system obtained by setting all variables to equal zero
- nontrivial solution
a solution to a homogeneous system that is not the trivial solution