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Question 34 - Week 4 Practice | Prasnya
Q34
00:00
25 Feb 2024
Q34.
Consider the vectors
v
1
=
(
1
,
−
1
,
0
)
v_1=(1,-1,0)
v
1
=
(
1
,
−
1
,
0
)
,
v
2
=
(
2
,
3
,
−
1
)
v_2=(2,3,-1)
v
2
=
(
2
,
3
,
−
1
)
, and
v
3
=
(
a
,
b
,
c
)
v_3=(a,b,c)
v
3
=
(
a
,
b
,
c
)
in
R
3
\mathbb{R}^3
R
3
. Choose the correct options from the following.
A
If
a
=
5
a=5
a
=
5
,
b
=
0
b=0
b
=
0
,
c
=
−
1
c=-1
c
=
−
1
, then the set
{
v
1
,
v
2
,
v
3
}
\{v_1,v_2,v_3\}
{
v
1
,
v
2
,
v
3
}
forms a basis for
R
3
\mathbb{R}^3
R
3
.
B
If
a
=
5
a=5
a
=
5
,
b
=
0
b=0
b
=
0
,
c
=
−
1
c=-1
c
=
−
1
, then the vectors
{
v
1
,
v
2
,
v
3
}
\{v_1,v_2,v_3\}
{
v
1
,
v
2
,
v
3
}
are linearly dependent.
C
If
a
=
5
a=5
a
=
5
,
b
=
0
b=0
b
=
0
,
c
=
−
1
c=-1
c
=
−
1
, and
A
A
A
is the matrix with
v
1
,
v
2
v_1,v_2
v
1
,
v
2
and
v
3
v_3
v
3
as its columns, then
rank
(
A
)
=
3
\operatorname{rank}(A)=3
rank
(
A
)
=
3
.
D
If
a
=
2
a=2
a
=
2
,
b
=
3
b=3
b
=
3
,
c
=
1
c=1
c
=
1
, then the subspace spanned by the vectors
{
v
1
,
v
2
,
v
3
}
\{v_1,v_2,v_3\}
{
v
1
,
v
2
,
v
3
}
has dimension
3
3
3
.
E
If
a
=
2
a=2
a
=
2
,
b
=
3
b=3
b
=
3
,
c
=
1
c=1
c
=
1
, and
A
A
A
is the matrix with
v
1
,
v
2
v_1,v_2
v
1
,
v
2
and
v
3
v_3
v
3
as its columns, then
A
A
A
is invertible.
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Check
Details
Q34
00:00
25 Feb 2024
Q34.
Consider the vectors
v
1
=
(
1
,
−
1
,
0
)
v_1=(1,-1,0)
v
1
=
(
1
,
−
1
,
0
)
,
v
2
=
(
2
,
3
,
−
1
)
v_2=(2,3,-1)
v
2
=
(
2
,
3
,
−
1
)
, and
v
3
=
(
a
,
b
,
c
)
v_3=(a,b,c)
v
3
=
(
a
,
b
,
c
)
in
R
3
\mathbb{R}^3
R
3
. Choose the correct options from the following.
A
If
a
=
5
a=5
a
=
5
,
b
=
0
b=0
b
=
0
,
c
=
−
1
c=-1
c
=
−
1
, then the set
{
v
1
,
v
2
,
v
3
}
\{v_1,v_2,v_3\}
{
v
1
,
v
2
,
v
3
}
forms a basis for
R
3
\mathbb{R}^3
R
3
.
B
If
a
=
5
a=5
a
=
5
,
b
=
0
b=0
b
=
0
,
c
=
−
1
c=-1
c
=
−
1
, then the vectors
{
v
1
,
v
2
,
v
3
}
\{v_1,v_2,v_3\}
{
v
1
,
v
2
,
v
3
}
are linearly dependent.
C
If
a
=
5
a=5
a
=
5
,
b
=
0
b=0
b
=
0
,
c
=
−
1
c=-1
c
=
−
1
, and
A
A
A
is the matrix with
v
1
,
v
2
v_1,v_2
v
1
,
v
2
and
v
3
v_3
v
3
as its columns, then
rank
(
A
)
=
3
\operatorname{rank}(A)=3
rank
(
A
)
=
3
.
D
If
a
=
2
a=2
a
=
2
,
b
=
3
b=3
b
=
3
,
c
=
1
c=1
c
=
1
, then the subspace spanned by the vectors
{
v
1
,
v
2
,
v
3
}
\{v_1,v_2,v_3\}
{
v
1
,
v
2
,
v
3
}
has dimension
3
3
3
.
E
If
a
=
2
a=2
a
=
2
,
b
=
3
b=3
b
=
3
,
c
=
1
c=1
c
=
1
, and
A
A
A
is the matrix with
v
1
,
v
2
v_1,v_2
v
1
,
v
2
and
v
3
v_3
v
3
as its columns, then
A
A
A
is invertible.
Save
Check
Details
