Linear Algebra 2270
Midterm 2
Group A
07/09/2015
Solve 4
1
Find
out
of 6 problems:
krne1 ;incl till range of till following lin ar transformation T :
ti1(
r’f/
.r
x
2
.t’
—*
1 + 2.r —
•r
)
=
J
I
2. Let U iw a suhspac(’ of F cl(’1un’d as lullows:
IV=
{ [‘]
c’
:.r
.
1
.r2
Ill
E
+.r2
Enrich a set
To a hasis of )j/.
.
)
Is there a Iin(ar t r nsfrinat ion T :
transiorinat ion exists, is it uinquv.’:
1’
([])
1. Prove the tlieoreiii that if T
5. Let I
h1
=
V
1 )( two sllhspa((’5 (If
using two V(’(t (irs
?Ii
e
1
TV
/2
[f].
—*
2 sueli that the following is satisfied?
F
T
([])
aboVe
a veutor spa(e
represeiltat 11111
Is lflHqlle
V
—+
.‘
([])
V Assiiiiie that all
!1I
+
=
[1]
V((t(
rs .r
112
ii amid I )lIly ii T1 fl Tl’2
II’. Write (lVtHhitIuIis ol
(a) T is linear
is
[]
2 as
E TV
6. Let 1. T4 is’ two v((’tor spaees amel T
(h) T
=
If sueh a
TV is 1—1. onto and imear, its iliverse is also lill(ar.
=
Prove that the
1
1-i
(u) I is onto
1
=
O}
may he written
rn
x
\J
o
o
•1
o
—
—
I
I
o
o
I
1-
—
0
I
-
4,
r’.
t
I’
0
‘I
—4
c---_____)
—
7Th
x.
1-
r----
()
3
c)
:3
-
O
r—
c.
—
I)
c
J2
4
c—.
I)
c
Q
L
ci
‘1
4
‘•x
(I
U
U
0
I
L
1
—
.zzz
(—I
—s
•—
—j’
-
L
ci
-
•E
-
4
tJ
—
0
I
S
-
N
—
-
—
0
—
c—;-----------
-—-----___i_
-
0
—
‘
0
Ci
-
CO
—
—
c
1-
ci
-
I-,
+
Il
r.
‘-%‘
r
,
j
-,
-
H
r—,
I,
-
j
I
I
r
-
(N
-
4-
c$__,
-‘
‘
3
1-
-
N
N
cc
cc
-4-
J
cc
II
cc
4
l
S
‘
2
C
0
1-
I’
S
-
N
-
n
--
0
0
I
—
4”
CC
4-,
N
t’__
‘-C
‘
.-c
‘4,
..z
N
—
—
p
cc
cc
Th
cc
-4-
c-c
—
c_c
—
t%4
Ij
-4’
cç
N
4’-
cc
4—
N
cr
cc
1
V
“3
—.4
N
(1
c
-S..’
cc
1
Lc
H
Lin’ar Algcl ua 2270
i\Ii(11(91I1 2
G1ouJ) 13
07/09/2015
Solvc 1 out
of
6
prob1enu-:
1. Let I V be a siihspace of fl defined as follows:
T1
=
{[]
e
2
,
1
:.r
,
.r
33 ..i
E
+,2
—
213
o}
—i
Enrich a set
To a basis of 117.
2. Find the kernel and the range of the fbllowiug linear transformation T :
2
/
f
L
3
J
([])
1. Prove the theorem that [
also lniv;ir.
: I/
=
.1
([])
is unique
(i Let V II Ia’ two vector spaces. Write
(h) V.IV
Ti
T
([h])
ii
Assmne
that all vectors
Y] + !12
and oniy if Wi fl l’V
definitions
of
If such a
=
V. L : 11, —k V are linear, then the
=
(a) (lHfl(/)
.13
[].
Let 1/Vj. I’V he two suhspaees of a vector space V.
lisin two vectors ?/l
lI i/i E W as
Prove that the above represelitatiomi
—
D such that the following is satisfied?
T
=
+ 2x
.11 + Zr2
/
3. Is there a linear trrnsformat ion T :
transformation exists. is it uinque?:
—
{}.
coniposit
.t
V
hoi ? a T
1 is
may
Ia’ written
_‘\j
S.—’
Si
“.4
2
“
cS
.
e:L
L
.0
3
--Th
I’
C’
5-.
-
L
—.,
-
:—
J
C,
-
cJ
‘-I
2
1
-1
-4-
V
.4-
z
c)
—
C.’
C
a