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Question 3 - Week 6 Practice | Prasnya
Q3
00:00
6 Apr 2026
Q3.
Let
T
:
R
3
→
R
3
T:\mathbb{R}^3\to\mathbb{R}^3
T
:
R
3
→
R
3
be defined by
T
(
x
,
y
,
z
)
=
(
x
+
y
,
y
+
z
,
z
+
x
+
k
y
)
T(x,y,z)=(x+y,\ y+z,\ z+x+ky)
T
(
x
,
y
,
z
)
=
(
x
+
y
,
y
+
z
,
z
+
x
+
k
y
)
, where
k
∈
R
k\in\mathbb{R}
k
∈
R
. Which statements are true?
A
T
T
T
is injective if and only if
k
≠
−
1
k\neq -1
k
=
−
1
.
B
dim
(
ker
(
T
)
)
=
1
\dim(\ker(T))=1
dim
(
ker
(
T
))
=
1
when
k
=
−
2
k=-2
k
=
−
2
.
C
The image of
T
T
T
is a plane in
R
3
\mathbb{R}^3
R
3
for all
k
k
k
.
D
The vector
(
1
,
−
1
,
1
)
(1,-1,1)
(
1
,
−
1
,
1
)
is in
ker
(
T
)
\ker(T)
ker
(
T
)
for some
k
k
k
.
E
There does not exist a non-zero vector
v
v
v
such that
T
(
v
)
=
v
T(v)=v
T
(
v
)
=
v
.
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Details
Q3
00:00
6 Apr 2026
Q3.
Let
T
:
R
3
→
R
3
T:\mathbb{R}^3\to\mathbb{R}^3
T
:
R
3
→
R
3
be defined by
T
(
x
,
y
,
z
)
=
(
x
+
y
,
y
+
z
,
z
+
x
+
k
y
)
T(x,y,z)=(x+y,\ y+z,\ z+x+ky)
T
(
x
,
y
,
z
)
=
(
x
+
y
,
y
+
z
,
z
+
x
+
k
y
)
, where
k
∈
R
k\in\mathbb{R}
k
∈
R
. Which statements are true?
A
T
T
T
is injective if and only if
k
≠
−
1
k\neq -1
k
=
−
1
.
B
dim
(
ker
(
T
)
)
=
1
\dim(\ker(T))=1
dim
(
ker
(
T
))
=
1
when
k
=
−
2
k=-2
k
=
−
2
.
C
The image of
T
T
T
is a plane in
R
3
\mathbb{R}^3
R
3
for all
k
k
k
.
D
The vector
(
1
,
−
1
,
1
)
(1,-1,1)
(
1
,
−
1
,
1
)
is in
ker
(
T
)
\ker(T)
ker
(
T
)
for some
k
k
k
.
E
There does not exist a non-zero vector
v
v
v
such that
T
(
v
)
=
v
T(v)=v
T
(
v
)
=
v
.
Save
Check
Details
