Lecture notes Mathematics II (MATH2011A) - ALGEBRA_Chapter_3
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Course
Mathematics II (MATH2011A)
Institution
University Of The Witwatersrand (wits)
This document clearly describes, with detailed notes and examples, how to evaluate/solve the following:
~ Linear Spaces
~ Bases
~ Dimensions
~ Independence
~ Rank
~ Eigenvalues and Eigenvectors
~ Diagonalization
~ The Characteristic Polynomial
as taught by the University of the Witwater...
HERE
,
{ XIY ;ZjW} ARE COMPLEX NUMBERS AND COULD CALL
THEM { Zijzz ;Z3;Z4 }
I WRITE DOWN AN AUGMENTED MATRIX :
CAN SEE HAVE 3 EQNS AND 4 UNKNOWNS i. MORE UNKNOWNS THAN EQUATIONS
IN SOLUTION (POSSIBLY MORE ) :
'
- -
EXPECT 1 PARAMETER
,
MATRIX THAT
q
% "°WS LEAVE ROW 9- AS IS : SINCE 9- AS FIRST ENTRY : IDENTITY MATRIX !
3i I 1- 3i
~
9- i 1 i I 1
n.eu#0-14ti1-i22iRz-iR12i-3-2tilti
i -2 Iti I 2 -
i
2 0 -
I 4 +i 1- i 2- Zi Rs -
2iR1
-
IN GAUSS JORDAN ELIMINATION :
GOAL IS TO PERFORM ROW OPERATIONS
UNTIL LHS LOOKS AS CLOSE AS
POSSIBLE 70 IDENTITY MATRIX !
* CAN ADD SUBTRACT TWO ROWS
OR MULTIPLY A ROW BY CONSTANT /
COMPLEX CONSTANT
!
Ng R
'S AFTER 1
moa-n.us JORDAN WANT ZERO
}
~
I 0 7- i -
I Zti 3 1- Zi , + i. Rz
CANT MAKE IT
MORE
look ANY MORE CAN MAKE
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THAN IDENTITY
☐
g-
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and
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UP
"
MATRIX ACHIEVED
O
WHAT WAS
O O O O R2 -
R3 ALREADY !
µgIN GAUSS JORDAN ELIMINATION :
GOAL IS TO PERFORM ROW OPERATIONS
UNTIL LHS LOOKS AS CLOSE AS
POSSIBLE 70 IDENTITY MATRIX !
2 i. AUGMENTED MATRIX REPRESENTS 3 EQUATIONS :
ox
toy 1- Oz tow =o
USEFUL EQUATIONS :
K + C- 1t7i)z + (zti )w = 3t2i
y
t C- 4- i)z + ( i 1) w -
= Zi -2
° ⇐ AND y) o
-5
VARIABLES :(ELIMINATION
WAY PERFORM Gauss
} USE TWO EQUATIONS TO ELIMINATE ANY
-
FROM
EASY TO ELIMINATE
EQUATIONS !
X
Y
✗ = 3t2i -
C- lt7i)z (zti )w
y = Zi -2 -
C- 4- iz -
( i 1) w -
, 4 THE GENERAL SOLUTION :
4 UNKNOWNS :
→ VECTOR
K 3t2i + (I -
7-i)z 1-
f- 2- i)w
y -2+2 i + ( 4+i)z + ( I -
i )w
= * sum
{
Z £
DO NOT
HAVE ANY 2- sqYI.net
OTHER EQNS > NUMBER
W W
'
- - CANNOT ELIMINATE
Z AND W
SUM CONSTANTS
ors
3t2i 1- 7- i -
2- I
= '
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+ z + w
0 I 0
0 0 9-
-1--0
| IN SOLUTION :
Z AND W
p CAN PUT SEPARATELY
INTO SYSTEMS OF EQNS
/ COMBINED
FROM CALCULUS : BECOME ANYTHING
ARBITRARY CONSTANTS
-
EG . 2C = ( I 7- i)z + ( 2-I)w
- -
IS PARTICULAR -
PARAMETERS CORRECT RHS
(
NOT
1
SOLUTION OF ORIGINAL y Rµg = 0 LIKE PARTICULAR SOLNO
SYSTEM OF EQUATIONS
( No ARBITRARY constants :)
To SOLVE HOMOGENEOUS EQUATION !
IF SUBSTITUTE THESE ELEMENT OF NULL
CONSTANTS INTO EQNS SPACE CEE
= RHS
°
GOING FORWARD
,
COMMON FOR EQUATION TO BE HOMOGENEOUS ONLY !
( PARTICULAR SOLN = 0 )
•
TWO WAYS TO INVERT A MATRIX :( INVERSE OF MATRIX
EXAMPLE :
I Iti 9-
FIND THE INVERSE OF
i
Iti o
i i Iti
1 PERFORM GAUSS -
JORDAN ELIMINATION OR
2 WORK OUT DETERMINANT AND ADJOINT
I 1 WRITE AUGMENTED MATRIX :
I 9- + i 9- I o o ~ I Iti 1 1 00
Iti i 0 0 TO 0 -
i -
I -
i -
I -
i g- o R2 -
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i i Iti oE t R3
- ¥ÉeÉ
O I I -
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-
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MATRIX WANT INVERSE
MATRIX ON RHS
OF ON LHS !
GAUS / AN ELIMINATION
-
. . PERFORM
TO GET IDENTITY MATRIX ON
LHS AND INVERSE ON RHS !
I 9- ti I l 00 Iti 1
~ ~ I 1 00
o I 1- i 1- i i 0 Rzci) o I I -
i 1- i i o
O I I -
I 0 I 0 0 I -
I -
i I Rs -
Rz
~ I Iti I l 0 O ~ I 9-
' '
0 Ricki
Iti Iti 0
O I 1- i 1- iio o I 1- i 1- iio
O O I I -
I -
i R3( i ) -
O O I I -
I -
I Tako
not use
# FRACTIONS !
GO COLUMN BY COLUMN
WITH 1 'S FIRST THEN
ZERO 's !
, ' l
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-
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i
→ it '
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~ 9- 0 /+ i
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i
i-zi-iz.li
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-
3- i i
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L
z prEFERABLEMET
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TO GET COFACTOR OF EACH ENTRY :
I +I
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9- 9- OUT ROW AND COLUMN OF POSITION IN
!
T
AND THEN WORK OUT DETERMINANT OF REMAINDER
'
1 A 2 iclti ) Ii)(Iti) E
i
adj A- ( Iti)
=
Iti 0 = -
i i Iti '
( Iti) -
i ( ai) i -
i -
( Hilli)
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i-
-
( Iti) i -
( Iti )
+
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t -
t
, iclti )
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2
adj A- = ( Iti) Ii)(lti) iz-
'
( Iti) -
i ( ai) i -
i -
( Hilli)
?
i -
-
( Iti) i -
( Iti )
ti -
l Zi + i T
adjA
-
-
i + I I -
+ -
i -
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l -
it i -
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HAVE
STILL
T
→
TAKE TRANSPOSE yo
I -
I -
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of This
MATRIX
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,
adj A =
-
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I
co -
EACH
FACTOR
ENTRY
OF
!
-
i 9- + i -
i
i -
i -
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i
%
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Zi 1 9- ti
i-1•
METHODS
OUT
TO
DETERMINANT
WORK
9- + i I
i'
I • '
3 det A =
det i
g- + i 0
i i Iti det =
iclti) to + iciti ) -
iz - o -
( Iti )
's
i Zi Citi)
=
i -
i + -
i ti -
SINCE KNOW ADJOINT OF A 8 Zi l Zi 2i2
3 = -
-
-
= 9-
•
def A = 1st ENTRY IN A adj A 2 COULD EXPAND ON Row /column : THAT HAS ZERO IN IT !
A adjA
✗
:±①
+ -
+
=
I 9- + i 9- i -
i i -
i Iti i ti
? Yi
-
=
+1 det - Odet + 9-
+ idet 1 '
= Iti i 0 -
Zi 1 Iti
i i Hii
i i Iti i -
I -
i = i -1+1 -0 + ( Iti ) ( i - zi )
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=
i + (i ziti
3×3 3×3
- -
=
it i -
Zi -1+2
i. 3×3 = 9-
9-( i )
•
=
i -
i + ( Iti )( -
Zi) t • First
ENTRY !
-
= i -
i -
Zi -
Zi + i
= 1
:
4 A
-
= adjA
detA
i i i
÷
-
i -
-
=
-
zi I 9- ti
i -
I -
i
÷ A- = i -
I -
i -
i
-
Zi 1 Iti
i -
I -
i
0
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