Lecture notes Linear Algebra (MAS2701)

, week Tuesday
24 9 24
. .




O Motivation


Linear
Algebra solving :
systems of linear equations

(*)
Ax = 1


A-MatCRI ,
bER


Est
Goal : Find all 1 = (holds




egA
(23) Mat (R) =
= -
,




1 X1 + 2x2 + 3X3 = 7



. x1
4 + 5x2 + 6x3 = 8 .




Goal :
Vast
generalization
1) more coefficients
general
*
al IR -
>
Vector spaces


3) matrices -
> linear maps


·
Develop a
theory
Gain
insight about matrices
·
more




Example : Let A distinct 13 EIR3.
be a 3x3 matrix with 3
eigenvalues 91 9293
, ,
and
corresponding eigenvectors 11 , 2 ,




for
Avizgivi i = 1
,
2
,
3




Let P = (11 : 12 : 13) be the matrix with columns1 ,
12, 13



Claim : Pisinvertible /-> need "linear independence").

, # Vector spaces




I 1 Fields of (complex) numbers


Goal : Develop
theory for Q ,R ,
C
simultaneously
Convention : All denote
theory MAS2701 the symbol If one of C , R, I



We callI f the field


Note : If has additionand multiplication
:




(i) Fa belf .
b
(commutativity)
a+ = b+ a
,




a .
b= b. a


(i) (a + b) +c = a+ (b + c) fa ,
b C +If
,
cassociativity).
(a b) - .
c = a (b -
c)


(iii) J 0. 1eFs t .
ota = a 1 a
.
= a Fatif .




(iv) Fatif Ibelf such that a+ b = 0




FatIFIEo3JcEF1Eo] such that al= 1



(we write b = -
a
,
c =
a) .




(v) a .
(b + c) = a .
b + a -

c fa b ceif
, ,




More a setIf (i)-(v)
generally ,
together
with two
binary operations + and on If such that holds is called a field


(Non) Example :




·
IN = Er ,
2,3 ...
] satisfies (i) (ii) but ,
not (iii) (iv).
,




2 E..;3 3 does cirl .
satisfy
·
= -3 -1 0 1 2 3
, ...
not
, , , , , ,




· In =
E 2 n-1 under +modnmodn
, , ...,




Zi0 50 99
eg n = 10 =
,
1,,
2
..,



In is field if and ifh is prime
a
only
(IR) does cil , i t
mat satisfy
·
not



(8)
g(86 (82) (8 )
%
e .
.


=
(8) = (8 % ) =
.




I . 2 The Definition of a vector space




)W
Recal : If =
,




=

, Definition : A field If in set
vector space over a a
together with two maps

+: VxV -
> V (a 1) <
a +r
, ,




· : FxV-> V
, (y 1)1,
>
you

hold
such that the
following :




1)u +1 =
1 + UF & Ver ( + is commutative

2) (a +1) +w = 4 + (+w) fU w + V (+ is associative
, ,




3)7 ger such that 1 +
0 =
- Ever(additive identity) .
4) Fuer Jer such that u +1 =
& ladditive inverse) .


5) 1 4 4 fe .

=




6) ) (. .
y (- Famef, multiplication
her of scalar
associativity
·
=




7) (y + H) fg M + F , Ye

3 distributivity
u .
·
=
y u + M4.


,




8) p (a+y you +y y
.

=
FyfF , Y Yer ,




Remark :

·
If If IR then we that V is a real rector space
=

say
We denote the additive inverse y w-l instead w + (4) .
of
by-u and we write
·




·
If If 1 then we that V is a complex rector
say space
=




Examples : 1) IR" is a real vector space


It is a complex vector space



If:al
I antify set of n-tuples of elements i a
in
general ,
as
, ..




is a vector space over If with component wise addition and scalar multiplication
2)
MatCF set of mxn matrices with entries in If .




MatHF) is a vector space over


3) If [x] = set of
polynomials with coefficients in IF




[ 3
*"
=
p(x) = anx"+ an-1X +... t dix + do : do , al, ..,
an elf



n+ 20 ,
1
,
2
,
...
3

If g(x) = bnx+... + bex + bo
*
(p + q)(x) = (an + bu)x + .. . + (a) + bi)x + ao + bo




(y-p((x) =
Gan)x" +... + (gai)x + yao .




we /F[x]XIFIx] >
If[X]
get maps +:
-


:




·: If x IF [X] > F[X]
-




5) 6) 7) 8) hold because
Conditions 11 ,
2)
, , , ,
they hold .
in If



3) Let g(x)
condition : = o zero
polynomial
Condition 4) : -
p(X) = -anx- ...
- a i x- a s