1
Lecture
Algebra
What
is
examples
algebra
and
first
addition
subtraction
3
operations
abstract
,
:
-
What do we think
about ?
numbers : IN , I, IR C Q
S
,
polynomials
Vector
objects
matrices
multiplication
spaces
division
factorization
:
--
Abstraction
What
,
are
I
identify and study the commonalities
:
commonalities
the
+o
·
of the
objects above ?
%
#
R
poly's
matrices/
&:
does the order of elements
th
of operations
matter ?
th ?
Determining which commonalities to focus
It takes time thought
studying
,
,
on
is
hand .
Why abstraction ?
clarify
examples
lead to
-
>
-
concrete
know
we
interesting examples
new
-
abstraction
example
eg
above examples
Definition :
A
-
lead to
together with
G
set
is a
grou
an
operation GXG + G
axf
>
-
:
(a , b)
satisfying
:
(1) associativity
a
"entity
JeeG
16x2)
=
Ja* b) * <
-
(ii)
such that
exa-a
(iii) Inverses
for any
A-G
Examples
:
(t
non-examples
,
+
)
,
,
there
such that
,
is
,
,
,
=
e = ada
,
,
(GLu/R) ,
(N ,t ) (1 ) ( *02 ·, 0
,
,
,
,
G
element
an
axa
at
(M +. (k + ) (@ , +) (Mn(R) + )
,
JISO3 , 1
:
G
as
for all
ax e
=
.
Eg I we'll be coming back to this in more detail soon)
>
X set
Sym() Ef X-X
bijection 3
=
with
cheek that
this
is
a
:
operation
group
fog composition