BASIC MATHEMATICS
.1
1
= Proofs [ > We often want to proove that a statement
is
-
case
↳ The induction
,
but for all
possible ones , thus we introduce:
-generally
use this one
k
3 steps !
-
=i
example :
Our
1) Base case >
1
=
13
↳
just use
=
a
your
n
,
lecture
General
axioms
3) Prov that statement holds for
nis =
notes =
simple number
x +
holds for some
work
(y z)
+
X +
y
=
=
y
(And of obvious)
(x y) zv (y z) (x y) z
+
.
+
x + 0 = 0 +x
x + (
n+ 1
-
x)
=
.
=
-
-
Y
+ x
0
↑
(*)
=
() x
.
=
1
part
i + (n + 1)
=
(i In
Cauchy-Schwarz
+
X
1
. 2 vectors > allows us to work
-
Triangle
Kv w) / Ell VIIII w/
.
-
: further
=
or
inequality
min
-(
by
is free
and find a contradiction
is false
example
choice
neN = A
-the actual
prove that a statement
we
assuming
12V holds
=
2) Assume that statement
k
-
prove that our statement
holds for a case of
k
by contradiction
↳ proof
principle
only applicable for one
not
expand both sides and
everything cancels out :
se en
Fideyusedin engreering,landescribe physical quantit
in multiple dimensions !
basis for many
aerospace systems
↳ Basic operations
·
v + w
=
nth dim.
rectors
4 Scalar
< a , b)
=
()() =
product
lallan
projection
ac by + azb2 +...
+
s
anbr
direction of b
.
↳ Cross product
neR3
nth dim.
=
,
=
+
·
=
11 a x b11
() (bab)- axb
↳ Mixed Product
[a b c]
11a b11 <11
,
< a , b c)
=
area
height
=
is
=
<ab.
<(a b)
,
< aib) + < 9 , c)
(a + b , a + b)
=
11911 + 24a , b) + 11311
< a b , a b)
=
11a(l
(a + b , a b)
=
<a a)
-
-
-
of a in
·
11 all 11 b1/ cos(2)
=
=
=
.
+
b
=
axb
<a , x b)
7a
a)
lall
< a , as =
-
,
orthe
alb
2(a , b) + 11311
=
-
<b , b)
gonality
<a , b) = 0
-area enclosed
119/111b1lsin(2)
orthogna o
volume enclosed
es
·
.2 1
1
.
Vector
Subspaces et of vectorsthaSharesomethingincombon
If
(V , +, )
i)
O f W-zerorector
is
W
w
iif
v
,
iii) FVEW
,
example1
a
rector space ,
=
v + we
GER
dvtW
=>
ab ctVV
=
,
i) (8) V
(X
W
Wis
3
a
>
cv
b
need to fulfill
if :
all of these
=
=> k = 0
,
=
+
,
it is
a, b , c
SX
vectors of
abe e en
EveR2/v (5) KERRY
,
=
V
i) (8) +V
a
are
(1) V WeV then v weV = v () , w ( )= V+ w (
iii) VeV , LER then a veVev (8) = X v ( * 4)
vecters in X direction
1Y
ye
subspace of V
length 1
=
a vector
=
.
)
+
=
/
space
12 (IIVI)
Ev
=
>
because IIV11
-
=
=
.
example 2
,
=
=
13
1X
not a
rector
space