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Week 4 Practice Questions | Prasnya
Q1
00:00
4 Aug 2024
Q1.
Two vector spaces are isomorphic if there exists an isomorphism between them. Which vector spaces are isomorphic to
R
2
\mathbb{R}^2
R
2
?
A
V
1
=
{
(
x
,
y
,
z
)
∈
R
3
∣
x
+
y
+
z
=
0
}
V_1=\{(x,y,z)\in\mathbb{R}^3\mid x+y+z=0\}
V
1
=
{(
x
,
y
,
z
)
∈
R
3
∣
x
+
y
+
z
=
0
}
.
B
V
2
=
span
{
(
1
,
0
,
1
)
,
(
1
,
1
,
0
)
,
(
0
,
1
,
1
)
}
V_2=\operatorname{span}\{(1,0,1),(1,1,0),(0,1,1)\}
V
2
=
span
{(
1
,
0
,
1
)
,
(
1
,
1
,
0
)
,
(
0
,
1
,
1
)}
.
C
V
3
=
{
[
a
b
c
d
]
∣
a
+
b
=
0
,
c
+
d
=
0
,
a
,
b
,
c
,
d
∈
R
}
V_3=\left\{\begin{bmatrix}a&b\\c&d\end{bmatrix}\mid a+b=0,\ c+d=0,\ a,b,c,d\in\mathbb{R}\right\}
V
3
=
{
[
a
c
b
d
]
∣
a
+
b
=
0
,
c
+
d
=
0
,
a
,
b
,
c
,
d
∈
R
}
.
D
V
4
=
{
x
∈
R
3
∣
A
x
=
0
,
A
=
[
2
−
1
1
−
1
−
7
1
1
2
0
]
}
V_4=\{x\in\mathbb{R}^3\mid Ax=0,\ A=\begin{bmatrix}2&-1&1\\-1&-7&1\\1&2&0\end{bmatrix}\}
V
4
=
{
x
∈
R
3
∣
A
x
=
0
,
A
=
2
−
1
1
−
1
−
7
2
1
1
0
}
.
E
V
5
V_5
V
5
is the column space of
[
6
−
3
2
1
]
\begin{bmatrix}6&-3\\2&1\end{bmatrix}
[
6
2
−
3
1
]
.
Save
Check
Details
Q1
00:00
4 Aug 2024
Q1.
Two vector spaces are isomorphic if there exists an isomorphism between them. Which vector spaces are isomorphic to
R
2
\mathbb{R}^2
R
2
?
A
V
1
=
{
(
x
,
y
,
z
)
∈
R
3
∣
x
+
y
+
z
=
0
}
V_1=\{(x,y,z)\in\mathbb{R}^3\mid x+y+z=0\}
V
1
=
{(
x
,
y
,
z
)
∈
R
3
∣
x
+
y
+
z
=
0
}
.
B
V
2
=
span
{
(
1
,
0
,
1
)
,
(
1
,
1
,
0
)
,
(
0
,
1
,
1
)
}
V_2=\operatorname{span}\{(1,0,1),(1,1,0),(0,1,1)\}
V
2
=
span
{(
1
,
0
,
1
)
,
(
1
,
1
,
0
)
,
(
0
,
1
,
1
)}
.
C
V
3
=
{
[
a
b
c
d
]
∣
a
+
b
=
0
,
c
+
d
=
0
,
a
,
b
,
c
,
d
∈
R
}
V_3=\left\{\begin{bmatrix}a&b\\c&d\end{bmatrix}\mid a+b=0,\ c+d=0,\ a,b,c,d\in\mathbb{R}\right\}
V
3
=
{
[
a
c
b
d
]
∣
a
+
b
=
0
,
c
+
d
=
0
,
a
,
b
,
c
,
d
∈
R
}
.
D
V
4
=
{
x
∈
R
3
∣
A
x
=
0
,
A
=
[
2
−
1
1
−
1
−
7
1
1
2
0
]
}
V_4=\{x\in\mathbb{R}^3\mid Ax=0,\ A=\begin{bmatrix}2&-1&1\\-1&-7&1\\1&2&0\end{bmatrix}\}
V
4
=
{
x
∈
R
3
∣
A
x
=
0
,
A
=
2
−
1
1
−
1
−
7
2
1
1
0
}
.
E
V
5
V_5
V
5
is the column space of
[
6
−
3
2
1
]
\begin{bmatrix}6&-3\\2&1\end{bmatrix}
[
6
2
−
3
1
]
.
Save
Check
Details
