I. 1
Systems of linear equations
A linear equation in the variables Xi .
.
. .
.
Xn is an equation that can be written
in the form dik t 92×2 t . . .
tanxn = b.
A
system of linear equations is a collection of one or more linear
linear equations involving the same Variables .
2×1 -
Xzt 1.5×3=0
A solution of the
system is a list (51,52 Sh ) Xi 4×3=-7
-
.
.
.
Of numbers that makes each equation a true statement when the values
51 , . . .
,
Sn are substituted for Xi , . . .
,Xn , respectively .
The set of all possible solutions is called the solution set of the linear
Two linear
system .
systems are called equivalent if
they have the same
solution set .
.
A system of linear equations is said to be consistent if it has either one
solution or
infinitely solutions is inconsistent if it has
many ; a system
no solutions .
The essential information of linear be recorded
a
system can compactly
in called matrix
a
rectangular array augmented
column
column
X, -
2×2 t X3 =
0 I -2 I row I -2 I 0
2×2 0×3 =
0 0 2 -0 0 2 -0 0
-
5×1 -
5×3 =
10 5 0 -5 5 0 -5 10
linear system coefficient matrix matrix
augmented
An
augmented matrix of a system consists of the coefficient matrix
with an added column containing the constants from the right side of the
equations
The size of a matrix tells how many rows and columns it has .
An m x n matrix contains m rows and n columns
The basic strategy to solve a linear system is to replace one system
with an equivalent system that is easier to solve .
Three basic to simplify linear
operations are used a
system :
1. replacement replace one row
by the sum of itself and a multiple
of another row
2. interchange interchange two rows
3. scaling multiply all entries in a row by a nonzero constant
, Two if there is of
matrices are called row
equivalent a sequence elementary
row operations that transform one matrix into the other
Two fundamental questions about a linear system are :
1. IS the system consistent ; that is, does at least one solution exist ?
2 .
If a solution exists ,
is it the only one ; that is ,
is the solution unique ?
1.2 row reduction and echelon forms
A nonzero row or column in a matrix means a row or column that contains
at least one nonzero entry .
A row)
leading entry refers to the leftmost nonzero entry ( in a nonzero
A
rectangular matrix is in echelon form if it has the following properties :
1 all nonzero rows are above rows of all zeros echelon matrix
.
any
2 -3 2 I
2. each leading entry of a row is in a column to the
right of the leading entry of the row above it 0 I -4 0
I
3 all entries in column below 0 0 0
.
a a
leading entry are zeros 2
If a matrix in echelon form satisfies the following additional conditions ,
then it is in reduced echelon form :
I 0 0
2g
4 the leading entry in each nonzero row is I
0 16
0 I
.
is in its column
5. each
leading 1 the only nonzero entry O 0 I 3
It a matrix A is now equivalent to an echelon matrix U ,
reduced echelon matrix
we call U an echelon form of A ;
if u is in reduced echelon form ,
we call U the reduced echelon form of # .
Uniqueness of the reduced echelon form
Each matrix is now equivalent to only one reduced echelon matrix .
A pivot position in a matrix A is a location in A that corresponds to a
leading 1 in the reduced echelon form of A. A pivot column is a column
of A that contains a pivot position .
A pivot is a nonzero number in a pivot position that is used as needed
to create zeros via row operations
pivot columns
*
I 4 5 -
g
-
7
am
* * *
0 4 form : 0
am
2 -6 -6 General * * *
*
pivot 0 0 0 0
am
0 0 -5 0
, row reduction algorithm to produce a matrix in echelon form :
This is
step I
begin with the leftmost nonzero column .
a pivot column .
The pivot position is at the top .
Step 2 select nonzero entry in the pivot column pivot If
a as
necessary
-
.
,
interchange rows to move this entry to the pivot position .
step 3 use row replacement operations to create zeros in all positions
below the pivot .
Step 4 cover cor ignore ) the row
containing the pivot position and
cover all rows ,
if
any ,
above it .
Apply steps I -3 to the
sub matrix that remains .
Repeat the process until there are no
more nonzero rows to
modify .
If we want the reduced echelon form ,
we perform one more step .
Step with the
5 beginning rightmost pivot and working upward and to
the left , create zeros above each pivot . If a pivot is not 1 ,
make it 1 by a scaling operation .
The combination of steps I -4 is called the forward phase of the row
reduction
algorithm . Step 5 is called the backward phase .
In the
following system of equations , the variables X, and Xz are called
basic variables and Xs is called a free variable .
The statement
"
Xz is free "
in the parametric description means that are free
you
to chose value for X3
any
{
.
I 0 -5 I ×,
-
5×3 = 1 Xi =
It 5×3
0 1 I 4 Xz TX3 = 4 Xz= 4 -
Xs
O O O O 0 =
0 Xz is free
whenever a system is inconsistent .
the solution set is empty , even
when the
system has free variables . In this case ,
the solution set
has no parametric representation .
Existence and uniqueness theorem
A linear consistent iff the rightmost column of the
system is augmented
matrix is not a
pivot column .
If a linear
system is consistent , then the
solution set contains either a unique solution or infinitely many
solutions c. when there is at least one free variable )
Systems of linear equations
A linear equation in the variables Xi .
.
. .
.
Xn is an equation that can be written
in the form dik t 92×2 t . . .
tanxn = b.
A
system of linear equations is a collection of one or more linear
linear equations involving the same Variables .
2×1 -
Xzt 1.5×3=0
A solution of the
system is a list (51,52 Sh ) Xi 4×3=-7
-
.
.
.
Of numbers that makes each equation a true statement when the values
51 , . . .
,
Sn are substituted for Xi , . . .
,Xn , respectively .
The set of all possible solutions is called the solution set of the linear
Two linear
system .
systems are called equivalent if
they have the same
solution set .
.
A system of linear equations is said to be consistent if it has either one
solution or
infinitely solutions is inconsistent if it has
many ; a system
no solutions .
The essential information of linear be recorded
a
system can compactly
in called matrix
a
rectangular array augmented
column
column
X, -
2×2 t X3 =
0 I -2 I row I -2 I 0
2×2 0×3 =
0 0 2 -0 0 2 -0 0
-
5×1 -
5×3 =
10 5 0 -5 5 0 -5 10
linear system coefficient matrix matrix
augmented
An
augmented matrix of a system consists of the coefficient matrix
with an added column containing the constants from the right side of the
equations
The size of a matrix tells how many rows and columns it has .
An m x n matrix contains m rows and n columns
The basic strategy to solve a linear system is to replace one system
with an equivalent system that is easier to solve .
Three basic to simplify linear
operations are used a
system :
1. replacement replace one row
by the sum of itself and a multiple
of another row
2. interchange interchange two rows
3. scaling multiply all entries in a row by a nonzero constant
, Two if there is of
matrices are called row
equivalent a sequence elementary
row operations that transform one matrix into the other
Two fundamental questions about a linear system are :
1. IS the system consistent ; that is, does at least one solution exist ?
2 .
If a solution exists ,
is it the only one ; that is ,
is the solution unique ?
1.2 row reduction and echelon forms
A nonzero row or column in a matrix means a row or column that contains
at least one nonzero entry .
A row)
leading entry refers to the leftmost nonzero entry ( in a nonzero
A
rectangular matrix is in echelon form if it has the following properties :
1 all nonzero rows are above rows of all zeros echelon matrix
.
any
2 -3 2 I
2. each leading entry of a row is in a column to the
right of the leading entry of the row above it 0 I -4 0
I
3 all entries in column below 0 0 0
.
a a
leading entry are zeros 2
If a matrix in echelon form satisfies the following additional conditions ,
then it is in reduced echelon form :
I 0 0
2g
4 the leading entry in each nonzero row is I
0 16
0 I
.
is in its column
5. each
leading 1 the only nonzero entry O 0 I 3
It a matrix A is now equivalent to an echelon matrix U ,
reduced echelon matrix
we call U an echelon form of A ;
if u is in reduced echelon form ,
we call U the reduced echelon form of # .
Uniqueness of the reduced echelon form
Each matrix is now equivalent to only one reduced echelon matrix .
A pivot position in a matrix A is a location in A that corresponds to a
leading 1 in the reduced echelon form of A. A pivot column is a column
of A that contains a pivot position .
A pivot is a nonzero number in a pivot position that is used as needed
to create zeros via row operations
pivot columns
*
I 4 5 -
g
-
7
am
* * *
0 4 form : 0
am
2 -6 -6 General * * *
*
pivot 0 0 0 0
am
0 0 -5 0
, row reduction algorithm to produce a matrix in echelon form :
This is
step I
begin with the leftmost nonzero column .
a pivot column .
The pivot position is at the top .
Step 2 select nonzero entry in the pivot column pivot If
a as
necessary
-
.
,
interchange rows to move this entry to the pivot position .
step 3 use row replacement operations to create zeros in all positions
below the pivot .
Step 4 cover cor ignore ) the row
containing the pivot position and
cover all rows ,
if
any ,
above it .
Apply steps I -3 to the
sub matrix that remains .
Repeat the process until there are no
more nonzero rows to
modify .
If we want the reduced echelon form ,
we perform one more step .
Step with the
5 beginning rightmost pivot and working upward and to
the left , create zeros above each pivot . If a pivot is not 1 ,
make it 1 by a scaling operation .
The combination of steps I -4 is called the forward phase of the row
reduction
algorithm . Step 5 is called the backward phase .
In the
following system of equations , the variables X, and Xz are called
basic variables and Xs is called a free variable .
The statement
"
Xz is free "
in the parametric description means that are free
you
to chose value for X3
any
{
.
I 0 -5 I ×,
-
5×3 = 1 Xi =
It 5×3
0 1 I 4 Xz TX3 = 4 Xz= 4 -
Xs
O O O O 0 =
0 Xz is free
whenever a system is inconsistent .
the solution set is empty , even
when the
system has free variables . In this case ,
the solution set
has no parametric representation .
Existence and uniqueness theorem
A linear consistent iff the rightmost column of the
system is augmented
matrix is not a
pivot column .
If a linear
system is consistent , then the
solution set contains either a unique solution or infinitely many
solutions c. when there is at least one free variable )
