Uploaded by Carson Kennedy Arias

HW4

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Igg
at Tca az s
1,9
vis
Not linear because
6 Tca and
a
property does nothold
at
taz 2am aa
Wts
ai
TL
II.lt
iiii 2la
ni
aatas
a
tail
Tai Tai
0
any
iii a safari
anais la
iii Iii
TCI
II
Tl
you
dependant
linearly
not 1.2
they are
Ic
2
I
C so it is not
linear
The two vectors
le
cannot pull out
D
D
we
are transforming are
after the transformation
T 11031
T1 2,0 6
1
C
2
I
Y
4
1 11
It
Kill
l'd
I
X E lil
T
Iif
If by
definition of
This
transformation mty
210
tank s Q
airstazuztasu t
Constiv transformation
t 9nTUnb
a Tcu tartly
a Tlv t
Then
0
CTCu
0
TLV title
Tatra
Q
Q
my linear
is
to
Q
b a Tcu tartly
N
t 9nTUnb
a Tlv t
a saz says an
Of
Contrapositive
y
eve or disprove
T Out
for any linear transformation Tiko W
Ow
is
on its thing
iii
of
7100
6
To
T owl
disprove this
consider
a linear transformation
8t Ids On
son did
linear transformation 71
the
transformation
Its
matrix is
1 11D
when you augment
t
a vow
of liras
the cull space
you can see that
is made up of
therefore u does not
have
to be Oy
I lot it
1
a
Eli
E Ii
6 tight
in
gp
a
i
iii
Rik
iii
columns have
The first two
the cult space
is spanned by
two s limns of A
spank
f
pivots so
So the
the first
basis
Ingot
Impt
I
iii
Ig
a
If
rites
I
i t t
E
vs
gg
lets
Null space
The busre to the
span
3
i
o
71
it
Flexi
Not
that
to
or so
s
Cx s
sit s ti
We have proven
TC rtv
Tcr
TLV
T
xxx
TC
T
first
same result
the
get
xp
This
transformations takes
ok
will
we
TW
ur
take the transformation
we
CTIA
yer
tallest to
CEE
stand ti
us to that
If
where
titz
it
Scaling a
or transformation
i ordinate
its x
I
titty
x
Till
TCal
CTW
1
Htt
Tal
xity
concludes the proof for showing
is a linen transformation
Pt 2
what
Prove that W
EPROI
It
Exert x
sacrifice the conditions to be in
Therefore
W S
Yet
only
I hit
the
Till 43
Latina
before
3
Let y be an arbitrary vector in the set
xeVIT
Y
is
x
some
Than I
vector
in
p
W
We
Exert Tai
Wu Lo u shown
Ye Ni
that yeExewlt.CH x
Thirtore
We know
x
EW
has shown both directions
concludes the
proof
This
we went to
Spell l
at
We know
independent
so
wÉ
they
are
I
are linearly
O vector
only share the
Éqval to R2
to have
basis for this space needs
independent vectors noticewhen
a
linearly
you
9
that
vectors they
the space they spacegenerate
in
two
for what
see
add
Wi
Wz youcombinethe span
Sp 91,1173
so our span becomes
This span has two C I D vectors
so
it equals R
then
b
find
intersect
a
shown
W
WE
formula T Ct yl where
is defined
as
in
T
the previous problem
9
b
that W
of problem
prove
Using the notation
T it now thatWa isthe
of
is the image
kernel of T
i
10
B be two matrices of sane size non
it A
thin
As
an
19
H
i
8
T IAI sans
Adat b
I
i
ant t th
n
fan
ATB
tannitylbl bi t
tbh
sa
t bi
Trial
Let s he
Trial
Tr
C
an
bin
and
I
ann
n
i
it 6mn
th
Ann t thn
t
Tr
t
i n
Cnnthin
01
thin
any constant
an
in
art
Fann
Tr
Al
set
fail
ant
a
n
Ann
At
fan
9in
any
Ann
i
T IA
fan t
forionvinae
let A
AO
0A
tann
let A
ED
B be 2 2
Ed
D
at tow
9,113
cx naw
I
E
Tv AB
att be
Tr COAL
x at
Tr AB
Tr la
text 2h
ya zt
Tr I A
Jw
matrix
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