Existence, Uniqueness, and Conditioning
Solving Linear Systems
Special Types of Linear Systems
Software for Linear Systems
Scientific Computing: An Introductory Survey
Chapter 2 – Systems of Linear Equations
Prof. Michael T. Heath
Department of Computer Science
University of Illinois at UrbanaChampaign
Copyright c
2002. Reproduction permitted
for noncommercial, educational use only.
Michael T. Heath
Scientific Computing
1 / 87
Existence, Uniqueness, and Conditioning
Solving Linear Systems
Special Types of Linear Systems
Software for Linear Systems
Outline
1
Existence, Uniqueness, and Conditioning
2
Solving Linear Systems
3
Special Types of Linear Systems
4
Software for Linear Systems
Michael T. Heath
Scientific Computing
2 / 87
Existence, Uniqueness, and Conditioning
Solving Linear Systems
Special Types of Linear Systems
Software for Linear Systems
Singularity and Nonsingularity
Norms
Condition Number
Error Bounds
Systems of Linear Equations
Given
m
×
n
matrix
A
and
m
vector
b
, find unknown
n
vector
x
satisfying
Ax
=
b
System of equations asks “Can
b
be expressed as linear
combination of columns of
A
?”
If so, coefficients of linear combination are given by
components of solution vector
x
Solution may or may not exist, and may or may not be
unique
For now, we consider only
square
case,
m
=
n
Michael T. Heath
Scientific Computing
3 / 87
Existence, Uniqueness, and Conditioning
Solving Linear Systems
Special Types of Linear Systems
Software for Linear Systems
Singularity and Nonsingularity
Norms
Condition Number
Error Bounds
Singularity and Nonsingularity
n
×
n
matrix
A
is
nonsingular
if it has any of following
equivalent properties
1
Inverse of
A
, denoted by
A

1
, exists
2
det(
A
) = 0
3
rank(
A
) =
n
4
For any vector
z
=
0
,
Az
=
0
Michael T. Heath
Scientific Computing
4 / 87
Existence, Uniqueness, and Conditioning
Solving Linear Systems
Special Types of Linear Systems
Software for Linear Systems
Singularity and Nonsingularity
Norms
Condition Number
Error Bounds
Existence and Uniqueness
Existence and uniqueness of solution to
Ax
=
b
depend
on whether
A
is singular or nonsingular
Can also depend on
b
, but only in singular case
If
b
∈
span
(
A
)
, system is
consistent
A
b
# solutions
nonsingular
arbitrary
one (unique)
singular
b
∈
span
(
A
)
infinitely many
singular
b
/
∈
span
(
A
)
none
Michael T. Heath
Scientific Computing
5 / 87
Existence, Uniqueness, and Conditioning
Solving Linear Systems
Special Types of Linear Systems
Software for Linear Systems
Singularity and Nonsingularity
Norms
Condition Number
Error Bounds
Geometric Interpretation
In two dimensions, each equation determines straight line
in plane
Solution is intersection point of two lines
If two straight lines are not parallel (nonsingular), then
intersection point is unique
If two straight lines are parallel (singular), then lines either
do not intersect (no solution) or else coincide (any point
along line is solution)
In higher dimensions, each equation determines
hyperplane; if matrix is nonsingular, intersection of
hyperplanes is unique solution
Michael T. Heath
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 Fall '08
 Layton,A
 Linear Systems, Matrices, Systems Of Linear Equations, Sula, Diagonal matrix, Conditioning Solving Linear Systems Special Types of Linear Systems Software for Linear Systems

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