Prasnya
Prasnya
Prasnya
Continue with Google
Question 20 - Week 6 Practice | Prasnya
Q20
00:00
1 Dec 2024
Q20.
Let
V
=
span
{
(
3
,
0
,
0
)
,
(
0
,
−
2
,
2
)
,
(
−
1
,
1
,
−
1
)
}
V=\operatorname{span}\{(3,0,0),(0,-2,2),(-1,1,-1)\}
V
=
span
{(
3
,
0
,
0
)
,
(
0
,
−
2
,
2
)
,
(
−
1
,
1
,
−
1
)}
. Let
T
:
V
→
R
2
T:V\to\mathbb{R}^2
T
:
V
→
R
2
be defined by
T
(
x
,
y
,
z
)
=
(
c
x
−
y
,
y
+
z
)
T(x,y,z)=(cx-y,y+z)
T
(
x
,
y
,
z
)
=
(
c
x
−
y
,
y
+
z
)
. Choose all correct options.
A
T
T
T
is not one-one for any value of
c
c
c
.
B
T
T
T
is one-one when
c
=
0
c=0
c
=
0
.
C
rank
(
T
)
=
1
\operatorname{rank}(T)=1
rank
(
T
)
=
1
for all values of
c
c
c
.
D
nullity
(
T
)
=
2
\operatorname{nullity}(T)=2
nullity
(
T
)
=
2
for some
c
c
c
.
E
There exist non-zero real numbers
α
\alpha
α
and
β
\beta
β
such that
α
(
3
,
0
,
0
)
+
β
(
0
,
−
2
,
2
)
∈
ker
(
T
)
\alpha(3,0,0)+\beta(0,-2,2)\in\ker(T)
α
(
3
,
0
,
0
)
+
β
(
0
,
−
2
,
2
)
∈
ker
(
T
)
.
Save
Check
Details
Q20
00:00
1 Dec 2024
Q20.
Let
V
=
span
{
(
3
,
0
,
0
)
,
(
0
,
−
2
,
2
)
,
(
−
1
,
1
,
−
1
)
}
V=\operatorname{span}\{(3,0,0),(0,-2,2),(-1,1,-1)\}
V
=
span
{(
3
,
0
,
0
)
,
(
0
,
−
2
,
2
)
,
(
−
1
,
1
,
−
1
)}
. Let
T
:
V
→
R
2
T:V\to\mathbb{R}^2
T
:
V
→
R
2
be defined by
T
(
x
,
y
,
z
)
=
(
c
x
−
y
,
y
+
z
)
T(x,y,z)=(cx-y,y+z)
T
(
x
,
y
,
z
)
=
(
c
x
−
y
,
y
+
z
)
. Choose all correct options.
A
T
T
T
is not one-one for any value of
c
c
c
.
B
T
T
T
is one-one when
c
=
0
c=0
c
=
0
.
C
rank
(
T
)
=
1
\operatorname{rank}(T)=1
rank
(
T
)
=
1
for all values of
c
c
c
.
D
nullity
(
T
)
=
2
\operatorname{nullity}(T)=2
nullity
(
T
)
=
2
for some
c
c
c
.
E
There exist non-zero real numbers
α
\alpha
α
and
β
\beta
β
such that
α
(
3
,
0
,
0
)
+
β
(
0
,
−
2
,
2
)
∈
ker
(
T
)
\alpha(3,0,0)+\beta(0,-2,2)\in\ker(T)
α
(
3
,
0
,
0
)
+
β
(
0
,
−
2
,
2
)
∈
ker
(
T
)
.
Save
Check
Details
