Lecture I
Defi
A liver is called
Eg .
system
X X
consistent if it has at least
one solution
3
>
No solution I solution
10
-
3 -
2
augmented
Eg
.
Triangular form
I 3
2
12
I
5
4
I matrix
S
X ,
-
34s
-
= 2 2
3X + X2 2X2] 2nd & 3rd
-
-
multiple 1st
2x , + 2x2 + X3 = 4 eliminate X ,
(iii) >
eliminate X c
(
I
-
&
S
-
2
J I 1)
J 8 -
7 -
14
,Lec2 Gaussian Elimination
Elimentary Raw Operations :
Eg 1)
Multiply a raw
by non zero
number
(i) R2 < R2-3R ,
2)
any
Substract/Add
row from any
a
multiple of
other row
3)
↓ 5
R3ER3-2R ,
swap any
I rows
S
Goal
!
(
1 o
-
> 2
-
:
017 I
Gaussian
S
* -A *
34 !
S
27 8
=
L
limination *
any
mem
.
-Da
↓
RztR3-2Rz 00D *
(S
-
7 x3 14
-
= -
Inconsistant formula
X3 = 2
If ever obtain 10 0 0 (a)
Keep going
, ...
↓
,
R3 = -
TRz (a70) ,
the formula is inconsistent
( :Ii (
2
I
3
-
3 -
"
Jef. Row Echelon form
~
↓
9 All zero are at the bottom
RR-4
,
5
(6
·
%
o
3
b) The
leading non zero
entry
of each
vow
(ot)
↓ Gaussian-Jordan is
Elimination
stricky to the
right of the
(PO)
leading nonzero entry of the
RERB row above (TEA Pirot e
Pivot 7/24)
S
x, =
4
8
(
* A D A A
S
Xz =
-
3
X3 = 2
g
① a *
so
o ↑
000010
Leading variables : The variables
associated to
pivot colums
Free Variables : - - non pivot colums
,Reduced Row Echelon form
1) In row echelon form
2) all pivots areIs
3) All Os are above pivots
in each colum O
( j
*
&
Free variables are associated
with infinite
mum of solutions
as
long as the
sys
is consistent
, Applications of Linear system
Homogeneous system
E
g Chemistry
.
.
all zeros on
right side
(. ( I -at least I solution : all o
* integrals
&
+= 4 get
(X ,X 2 Xa
, , 4) =
( . 5 3
, , 4)
&
43
= t
x4 = t
CoHioNyOz + 02 -
> CO2 t Hag
+ NOz
Eg
. Traffic flow X6
* >
X4
3
*T
T
~
↑2
x5
7
13
x2+
x, X4
=
* 2tXz =
X5
x4 + x5 =
Xo + X7
↓ + X3 :
NotXn
S
1 + 3
I ·(
-
1000
S 110-100 D
000/ 1 +
1-1 J
10100 1-1 -
Defi
A liver is called
Eg .
system
X X
consistent if it has at least
one solution
3
>
No solution I solution
10
-
3 -
2
augmented
Eg
.
Triangular form
I 3
2
12
I
5
4
I matrix
S
X ,
-
34s
-
= 2 2
3X + X2 2X2] 2nd & 3rd
-
-
multiple 1st
2x , + 2x2 + X3 = 4 eliminate X ,
(iii) >
eliminate X c
(
I
-
&
S
-
2
J I 1)
J 8 -
7 -
14
,Lec2 Gaussian Elimination
Elimentary Raw Operations :
Eg 1)
Multiply a raw
by non zero
number
(i) R2 < R2-3R ,
2)
any
Substract/Add
row from any
a
multiple of
other row
3)
↓ 5
R3ER3-2R ,
swap any
I rows
S
Goal
!
(
1 o
-
> 2
-
:
017 I
Gaussian
S
* -A *
34 !
S
27 8
=
L
limination *
any
mem
.
-Da
↓
RztR3-2Rz 00D *
(S
-
7 x3 14
-
= -
Inconsistant formula
X3 = 2
If ever obtain 10 0 0 (a)
Keep going
, ...
↓
,
R3 = -
TRz (a70) ,
the formula is inconsistent
( :Ii (
2
I
3
-
3 -
"
Jef. Row Echelon form
~
↓
9 All zero are at the bottom
RR-4
,
5
(6
·
%
o
3
b) The
leading non zero
entry
of each
vow
(ot)
↓ Gaussian-Jordan is
Elimination
stricky to the
right of the
(PO)
leading nonzero entry of the
RERB row above (TEA Pirot e
Pivot 7/24)
S
x, =
4
8
(
* A D A A
S
Xz =
-
3
X3 = 2
g
① a *
so
o ↑
000010
Leading variables : The variables
associated to
pivot colums
Free Variables : - - non pivot colums
,Reduced Row Echelon form
1) In row echelon form
2) all pivots areIs
3) All Os are above pivots
in each colum O
( j
*
&
Free variables are associated
with infinite
mum of solutions
as
long as the
sys
is consistent
, Applications of Linear system
Homogeneous system
E
g Chemistry
.
.
all zeros on
right side
(. ( I -at least I solution : all o
* integrals
&
+= 4 get
(X ,X 2 Xa
, , 4) =
( . 5 3
, , 4)
&
43
= t
x4 = t
CoHioNyOz + 02 -
> CO2 t Hag
+ NOz
Eg
. Traffic flow X6
* >
X4
3
*T
T
~
↑2
x5
7
13
x2+
x, X4
=
* 2tXz =
X5
x4 + x5 =
Xo + X7
↓ + X3 :
NotXn
S
1 + 3
I ·(
-
1000
S 110-100 D
000/ 1 +
1-1 J
10100 1-1 -
