Question 34 - Week 4 Practice | Prasnya
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Week 4
Question 34
00:00
Est. 3 min
Marks:
+3.00
0.00
Consider the vectors
v
1
=
(
1
,
−
1
,
0
)
v_1=(1,-1,0)
,
v
2
=
(
2
,
3
,
−
1
)
v_2=(2,3,-1)
, and
v
3
=
(
a
,
b
,
c
)
v_3=(a,b,c)
in
R
3
\mathbb{R}^3
. Choose the correct options from the following.
Multiple correct
Medium Difficulty
25 February 2024
A
If
a
=
5
a=5
,
b
=
0
b=0
,
c
=
−
1
c=-1
, then the set
{
v
1
,
v
2
,
v
3
}
\{v_1,v_2,v_3\}
forms a basis for
R
3
\mathbb{R}^3
.
B
If
a
=
5
a=5
,
b
=
0
b=0
,
c
=
−
1
c=-1
, then the vectors
{
v
1
,
v
2
,
v
3
}
\{v_1,v_2,v_3\}
are linearly dependent.
C
If
a
=
5
a=5
,
b
=
0
b=0
,
c
=
−
1
c=-1
, and
A
A
is the matrix with
v
1
,
v
2
v_1,v_2
and
v
3
v_3
as its columns, then
rank
(
A
)
=
3
\operatorname{rank}(A)=3
.
D
If
a
=
2
a=2
,
b
=
3
b=3
,
c
=
1
c=1
, then the subspace spanned by the vectors
{
v
1
,
v
2
,
v
3
}
\{v_1,v_2,v_3\}
has dimension
3
3
.
E
If
a
=
2
a=2
,
b
=
3
b=3
,
c
=
1
c=1
, and
A
A
is the matrix with
v
1
,
v
2
v_1,v_2
and
v
3
v_3
as its columns, then
A
A
is invertible.
