Linear algebra assignment Vil
Amar O'Egirer CHEF'T
17302016004
page 147
Find the rank of the following vector groups and one of their maximum linear
independent groups, and express the remaining vectors with the maximum linear
independent group linearity:
ii
fo 00 if
3
to
Since there are
3
non zero rows
9,92 as are maximum
9g 39 15oz Az
E G
Ey G
if
E and Ez
Cc and ca not being
E
and
Ez
0
Ez
3
independent subsets
3,1 2
Eg 12,1 5,6
2,4
1
2,0
prove that
i
Ez
1
rank
3
0,7 14
are linearly independent
are linearly dependant
then
allzero
we have
using basic equation
and as there are no solution
C and Cz are linearly Zandt
4
equation
for
c
E the 2 0
for Gando
those equations
c
Ci
Rank S
E Ez Ez
are
maximum
linearly
subsets
independent
15
prove
g
8
R
RCB
r
Rft8
troops
R A RCB
2 prove
Ctf
r
zrCA
Her
therefore
r
B
tr
8D
d
7
RCAltRCB
R A t RCB
page 148
we suppose
r
r
a
AB
En
and
Emin
AB em
r
Ab En
rfa
r B
so
m
non zero
rft
rfAB En
and
AB
em
matrix
It
B em
therefore
when
nem
must have
x 0
solution
O
let It is mxn
men and RCA
is non B matrix
AB Im
counterig9IB
r
rant
f
if there
and
RFA en
Knowing
and r B
matrix
mxm
column vector
m dimensional
is
and
AB is
that
Im
ther
r
then
AB
AB
RCI
should
m
be
m
and because
rfAB
rfA
is man matrix men
m
prove
Mem
that
page 151
5
Clams beggz.az
ay
we know that
independent
it
if
means
which
7
can
a
cannot
a
be
ae as
as
expressed
a
be
can
that it
expressed
by
a
also can be expressed
is impossible as az ay ay linearly
cannot be expressed
therefore an
as
r
Therefore
Ha 3
are
linearly
as as
be
AB
r
Ij
therefore ne r B
as B has n columns
so
Bi
a
and as
too
correalated
linearly independent
are linarly
ae
are
B's
and
r B
column
ai
132
n andbecause
AB
Bo
by
ar as
independant
by
e r
9,929
B
B En
n
vector group is
as
4
r
r
ae 93
linearly
indep
C
a
and a
are
therefore
rank is
linearly
are
a az as
I
a
a
linearly
are linearly
o
therefore
a
as
thus
As
can
it also
Therefore
13 b
30 f
be
Correalated
expressed
3b
by
can
be
a
az Bsg is linearly
expressed
2b 10
O
0
2b lo
t
9 873
a
independent
also have rank
2
a
3A 12A
carrealated
Ba B
13
and
so Bi Ba 133
and
the max linearly
are
that
and now we know
correlated
a
by
as as
a oh
correlated
page 152
56
l
II
ftp.IBf
1 71
HE
ABI
E
ABI
IE El HIEI
fIIBI
prove
ftp.ABE
fIpgttBEl
ABI
A I BA
A AB A
KI AB At
fI
II BAI
II ABI
FI ABI A
H ABI
II
1 ALI BAH
I Al II ABI
E ABI
ABI
IAI
IAI II BAI
IAI II BAI
II BAI
9
detchI BA
if N is eigenvalue of AB ABY dX x fo
and by multiplying two sides
by B we
prove
detfxI
will get
AB
BA Bx
and we can
eigenvalue
of BA
X Bx
it is also
see that
Therefore
KI
ABI
an
LAI BAI