25- zw
dw >
-
cr
+
pill my
plane .
=
(g)
-1
.
1) 1010 (5
>
-
u
18]
=
a
+v +
11411
11211
1 I wil
w
v
=
=
5
12
81 +
=
,
=
Votto
=
,
(5]
[, ]
=
o
10 , 12)
,
w
=
.
< i2] .
2)
[ = ][]
An =
linear
·
2c
-
.
[ _, ] d(z] e[i] (b)
+
-
d
-
-
2c
d
ze
2x0
-
d
= 0
-
.
1
=
-
d
3d
2d
=
-
= 0
e
-
+
+
1
=
+ 2d
c
-
1
comb
[i]
=
+
-
ye
ze
=
=
0
=
1
2e
2
=
=
0
2
Ej
.
2c
-
1
=
d =
2 + ze
2
=
=
1
2c
2
=
1
+
7
z
Et
Au
[J] [b] .
=
=
3)
a)
[i]
v =
,
w =
%]
= =
cost
V w =
.
1
.
1
+
3
-
0
E
=
coSO =
-
O
60
/
=
b)
v =
[2 ]
w =
,
,
v
.
w
n
=
LOS
-
2
v-w
-
11V11IWIl
:
0
=
90
-
2
=
(z]
=
O
0
=
x)
[i]
=
v
v w
-
3
1 +
11 VII
=
2
11211
=
2
a)
(3]
v =
vw = y
-
1 (v11 :
ITo
11 wil
15
=
2
w =
,
2
=
[ii]
Cos
.
= =
%
0
w
,
=
-
=
=
68
[ = z]
3
=
Ft5
=
E
=
--1
:
0
=
135
X2
o
30
US
60
go
·
90 + 43
=
195
o
-
13
1s +
=
28+ 1
u)
when
11
are
if
9)
v
u
.
u
V
.
u
u
V
.
b)
u v
.
u
=
I
St
V,
-
V =
=
+ 1
0
1
1
-
V ,
-
w
1
4
=
= 0
and
=
114 11
=
u
u =
v
False
·
(V +2 w)
.
+
0
=
=
0
=
=
10 , 1
.
,
- 1]
CW
they are
+ 2w
ur +zU.w
u
w
,
:
u>
0
.
True
.
VIIw
,
v=
u
:
u
W
0
0
2)
.
(1 , 0 , -)
,
= 0
w
.
they
then
= 0
10
-
n
.>
1)
.
en.
=
w
.
another
0
=
w
counter
= 0
w
=
(11
=
u
multiple of
one vector is
.
not
11
.
-
0
0
.
-
v
unit
I
(4 , 42 43)
,
·
unFus
-Tur
+
11411h
(41
,
42 ,
43)
=
v.2 + up + us
#K-r11
=
11411 + 11
both
unit
are
1 + 1
...
.
114-V11 = 2
~
5)
A
2
=
vectors
[3997]
=
↓
True.
,
B:
/12220 3
>
-
Y
Rank = 1
space
6)
-
independent
·
Col
A
(A) =
=
-
(2c
=
2C
-
-
IA1
=
3
-
2
o
=
3
-
C
-
4
c + 3
#3
.
indrow
>
[in J
1
1
.
det A = 0
=
↓
Runk =
col-space +
row
↓
↓
↓
>
-
.
a)
-
-
1(3C
b) + 7
-
7) +
=
)
dependent
col
B =
[ 100 3
=
1(1)
=
1
(B1
+
-
0
+
(1)
C
#-
[J
=
(C)
=
<
=
-
=
-
·
d
=
(b
g(
-
+
1S)
3
-
c(12
+
34
-
3
Fo
[ii]
=
S
1S) + 2(6
-
3)
7)
A =
c
[22]
,
B
=J
[v]
CR
=
identify .
be
J
[%
R =
B =
C
=
Cr
.
=
S
C
-
B =
R
.
[]
=
R Should
]
[
1222]
22
2
O
c
Y
U
O
001
S ][00]
=
1
1
o
un
0
0
G
0
=
to
0
o
1
0
6
0
200
111
=
11
-
210
006
O
100
200
o
1
I
o
o
6
-
110
U
000
n
006
o
I
o
21
22
006
o I
220
=
2
onn
8
00
222
6
0
1
o
6
o
- =
110
00
o
0
h
b
8)
if
Al = 0
pivots
A =
>
-
.
elimination
[2j
: elimination
a
[
=
a(a
-
=
0
-
a3
-
-
or
0
a = 2
.
.
1A1
z(az
0
-
=
Ga + 8)
u)(a
,
-
na) +
2a2 + 89
6a2 + 8a
-
=
=
=
0
.
a d
a
a(az
a
((a
a
2)
aJ
na2
-
a
will fail
(a2 -na)
ab
give
a -2
=
=
A
to
-
=
9)
Fail
4
,
-
2
= 0
0
= 0
2)
=
0
paaz)
-
10)
9)
uxh
->
000
o
oo
·
0
o
d
000
O
o
0
b)
zeros
in
how
(A)
0
True
=
1
1
=
=
1
-
=
.
is
(A)
0
=
not
·
0
.
11
Fase
C)
-
A-invertible
:
AA- =
det
(AA
det A
.
IAI + 0
I
)
=
/
=
:
/
-
det A-
At
in
.
det(1)
detA-1 =
det =
·:
-1
.
= 0
dest A + 0
.
invertible
.
.
A2
det [A21
det A
=
det Ar
det A
.
.
(detA12
=
detA
+ 0
.
1: A isiur
det 12 0
invertible
· A2
.
True -
(1)
A
AA"
AA"
=
uvT
I
=
AA"
=
=
=
I
I
(I
-
I
v Tu)
+ urT
+
urT
A
=
1 +
ur (I
when
Tu =
-
vTu)
+
1.
.
(I- urT)
I
,
Au = o
.,
(I - urT)
=
A"
1
=
-
.
"
If
+
ur (E vius
=
urT(E
(E-vTu)"
-
v Tui
I
+
-
ULTI + urT (I -VTu)
(I-rTus"turTCEvruiturt
12)
EA =
U
.
[]
A =
[] 3
8
[] 100
=
E
E
=%
[ ]
EE"
=
E- =
Find the inverse OFE
.
I
EF
=
[00] /
[i] []
E
=
Lu
=
[i]
=
1
[i] So J
n
=
2/
8
h
3
633
A
2
=
.
=
A
2
AT
13)
-
> ex. rows and
A
[i]
·
AT =
=
A
col
AT
=
-
-
[A]
: A
= At
symmetric
False .
b)
A
=
AT
(ABJT
,
=
B
BT
=
BTAT
AB
=
BA
-
-
-
(AT B )T =
+
AB
- Cbe
A B
=
BA
+
re
needs
False
C)
A = AT
ret
1
A=
=
+
then
[i]
T
At A
AT = [t]
i (Aadijt-
-
.
(A 1995
A
+
=
=
+ [ -)
=
A
23] [i ]
*
+
IAF0
[ =]
:
A
:
[ + j []
-
is invertible
At
is symm
.
=
:
(A-T
AT
#
not symmetric
:
True -
d)
A , B ,
A = AT ,
C ,
symet
B = BT
(ABCIT =
True .
C
+
,
c =
BTAT
(ABCIT
.
CT
=
c
BA
-
=
LBA
.
.
stat atc
,
T
a
an
E ,
c-orthogonal
mattis
unit
a) a--an
G
=
I
=
<an
...
]
=
i
qTqn
T
92
b)
a , Tar
C)
Q11
=
Rotation
=
92
=
[0; ]
: unit
1
=
0
and
gm
-
daj
=
.
0
Cost
mattin
=
coso-sing
[]
sing
LOS