Lecture notes Mathematics II (MATH2011A) - ALGEBRA_Chapter_3
47 views 1 purchase
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
0 I -4 j -
j -
f 2 I -2
-
R2
THAN IDENTITY
☐
g-
But
and
messes
o.si .
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
= '
-2+21 4ti 1- i
+ 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 -
Citi)R1
i i Iti oE t R3
- ¥ÉeÉ
O I I -
I 0 I
-
IRI
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
R1(¥+
-
Éii
'
i
→ it '
I 1
I
~ 9- 0 /+ i
-
0 R, -
Rz ~ i -
ai 0
iii. ,
0 I 1- i 1- ii o o I -
i 1- ii o
o o 1 i -
i -
i 0 0 1 i -
i -
i
? Iai ? %
-
~ I 0 0 i Ri -
R}
~
I 0 0 ' i
o I I -
i 1- i i o o I I 9- Ii 0 Rz(
0 0 1 i -
i -
i o o 1 i -
i -
i
i
i-zi-iz.li
-
3-
~
I 0 0
0 I 0 1- iii. i Rz -
Rs
o o 1 i -
i -
i
-
3- i i
1- Zi 1- zi I "
W"
i. INVERSE : !
1- i ¥ i approach
fractions
i -
i -
i no
L
z prEFERABLEMET
"Ñi^=detA%d¥a
:
¥É=
TO GET COFACTOR OF EACH ENTRY :
I +I
# BLOCK
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)
'
i-
-
( Iti) i -
( Iti )
+
I t
-
t -
t -
t
, iclti )
' T
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 -
-
l -
it i -
:} ¥ÉE%umNs!
"
HAVE
STILL
T
→
TAKE TRANSPOSE yo
I -
I -
2J [ TO
of This
MATRIX
g,,
n.gg , ,n
,
adj A =
-
[ I -
I
co -
EACH
FACTOR
ENTRY
OF
!
-
i 9- + i -
i
i -
i -
i -
i
%
ddjA = -
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 )
ziz )
'
=
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
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller Akshay101. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $15.64. You're not tied to anything after your purchase.
