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May 22nd

May 22nd

# Linear Equations in Linear Algebra

## 1.1 Systems of Linear Equation s

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

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