• Equivalent Systems are linear systems that have the same solution set. It is
the book if two inconsistent systems will automatically be equivalent since both
solution (solution set is empty). However, by the de¯nition of row equivalent
(see below), the
two inconsistent systems are not automatically `row equivalent'. Nonetheless,
is de¯ned for matrices, not a linear system itself.
• Row Equivalent: If there is a sequence elementary row ope rations that
transform one matrix
into another, the two matrices are `row equivalent'.
• Consistent System
• Inconsistent System can be shown by reduced
echelon form. If the leading column is the
augmented row, the system is inconsistent. Note, in reality , there is no need to
echelon to prove inconsistency, but reduced echelon form does provide a solid
• Existence and Uniqueness : In augmented matrix, if we can arrange a matrix so
row has non zero value in the last column, but zero values for any other columns
in the same
row, the linear system is inconsistent.
1.2 Row Reduction and Echelon Forms
• leading entry of a row is the leftmost nonzero entry in a nonzero entry.
• pivot position in a matrix A is a position corresponding to a leading 1 in an
form of A. (We include a matrix A in de¯nition since the pivot position depends
to matrix. In such a case, inclusion of a speci¯c sample will make de¯nition
• pivot column is a column corresponding to a pivot position. More formally, it
sponding to a leading column in reduced echelon form. Note that the reduced
is needed since each matrix has exactly one reduced echelon form.
• forward phase is a phase to create a legal leading entry for echelon form. It
`forward phase' because zero entries under the leading entry will be created.
This is opposite
to backward phase in that backward phase creates zero entry above the leading
• backward phase is a phase of creating zero entries above
a leading entry to build a reduced
echelon form. For non-reduced echelon form, this phase is not needed.
• partial pivoting is a technique done in forward phase to reduce roundo® errors
the biggest value as a leading entry. To build an echelon or reduced echelon
this technique is not required.