• Linear Equation is an equation that all variables
have degree 1 and n one of them multiply
each other. An equation with a term such as x1x2 or x2 is not counted as a
linear equation.
• System of Linear Equation or Linear System: Linear system is a collection of
linear
equations involving the same variables. Remember that those equations must
involve
the same variables.
• Solution of a Linear System is a list (s1,s2
,…,sn) of numbers that makes
each equation a
true statement when the numbers (s1, s2,…, sn) are substituted for variables
x1, x2,…,xn in
a linear system.
• Solution Set
• Equivalent Systems are linear systems that have the same solution set. It is
unclear from
the book if two inconsistent systems will automatically be equivalent since both
have no
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,
`row equivalent'
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
use reduced
echelon to prove inconsistency, but reduced echelon form does provide a solid
framework for
proof.
• Matrix for a Linear System
• Coefficient Matrix or Matrix of Coefficients
•Augmented Matrix
• Matrix Size
• Elementary Row Operations
• Property about solution set and equivalency
• Existence and Uniqueness : In augmented matrix, if we can arrange a matrix so
that one
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
reduced echelon
form of A. (We include a matrix A in de¯nition since the pivot position depends
on matrix
to matrix. In such a case, inclusion of a speci¯c sample will make de¯nition
more solid.)
• pivot column is a column corresponding to a pivot position. More formally, it
is corre-
sponding to a leading column in reduced echelon form. Note that the reduced
echelon form
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
is called
`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
entry.
• 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
by selecting
the biggest value as a leading entry. To build an echelon or reduced echelon
form, however,
this technique is not required.
• free variable
• basis variable
References
