Question 18 - Week 3 Practice | Prasnya
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Week 3
Question 18
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Est. 3 min
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Unless otherwise stated, assume that we consider the usual vector addition and scalar multiplication. Choose the correct option(s) from the following statements.
Multiple correct
Hard Difficulty
29 October 2023
A
The set of vectors
{
(
a
,
b
)
∈
R
2
}
\{(a,b)\in\mathbb{R}^2\}
with scalar multiplication defined by
k
(
a
,
b
)
=
(
0
,
k
b
)
k(a,b)=(0,kb)
forms a vector space.
B
The set of real numbers with addition defined by
x
+
y
:
=
x
−
y
x+y:=x-y
forms a vector space.
C
Let
A
∈
R
3
×
3
A\in\mathbb{R}^{3\times3}
be an invertible matrix. The set of all solutions of the homogeneous system
A
X
=
0
AX=0
is a vector space of dimension
0
0
.
D
Any vector subspace of
R
2
\mathbb{R}^2
with dimension
1
1
is of the form
a
x
+
b
y
=
0
ax+by=0
where
a
≠
0
a\ne0
or
b
≠
0
b\ne0
.
E
Any vector subspace of
R
3
\mathbb{R}^3
with dimension
1
1
is of the form
a
x
+
b
y
=
c
ax+by=c
where
a
≠
0
a\ne0
or
b
≠
0
b\ne0
.
F
V
=
{
[
a
b
c
d
]
:
a
+
b
=
c
+
d
}
V=\left\{\begin{bmatrix}a&b\\c&d\end{bmatrix}:a+b=c+d\right\}
forms a vector space. (G) The set of all
n
×
n
n\times n
matrices with rank strictly less than
n
n
does not form a vector space.
