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Practice Question: Subspaces
Related Topics
Wize University Linear Algebra Textbook > Vector Spaces and Subspaces
Subspaces
3 Activities
For the following subset
W
W
W
of
R
4
\mathbb{R}^4
R
4
, select the statements which are true:
W
=
{
w
⃗
∈
R
4
∣
A
w
⃗
=
5
w
⃗
}
W=\{\vec w\in \mathbb{R}^4\;|\;A\vec w = 5\vec w\}
W
=
{
w
∈
R
4
∣
A
w
=
5
w
}
where
A
A
A
is a
4
×
4
4\times 4
4
×
4
matrix
W
W
W
is a subspace of
R
4
\mathbb{R}^4
R
4
W
W
W
is closed under addition but not closed under scalar multiplication
W
W
W
is closed under scalar multiplication but not closed under addition
W
W
W
is not closed under scalar multiplication or addition
I don't know
Check Submission
More Subspaces Questions:
Span
Practice: Spans are Subspaces
Suppose that
v
⃗
1
,
…
,
v
⃗
k
∈
R
n
\vec v_1, \dots, \vec v_k\in\reals^n
v
1
,
…
,
v
k
∈
R
n
. Prove that
U
=
span
(
{
v
⃗
1
,
.
.
.
,
v
⃗
k
}
)
U=\operatorname{span}(\{\vec v_1, ...,\vec v_k\})
U
=
span
({
v
1
,
...
,
v
k
})
is a
subspace
of
R
n
\mathbb{R}^n
R
n
.
Subspaces
Practice: Subspaces
Let
M
2
×
2
M_{2\times2}
M
2
×
2
be the vector space whose vectors are
2
×
2
2\times 2
2
×
2
matrices with real number entries, and with the standard operations of matrix addition and scalar multiplication.
Consider the following subset of
M
2
×
2
M_{2\times2}
M
2
×
2
:
Show that a subset is a subspace
Let
V
=
{
(
x
,
y
)
∣
x
∈
R
,
y
∈
R
+
}
V=\{(x,y)\;|\;x\in\mathbb{R},y\in\mathbb{R}^+\}
V
=
{(
x
,
y
)
∣
x
∈
R
,
y
∈
R
+
}
be the vector space with operations:
(
x
1
,
y
1
)
+
(
x
2
,
y
2
)
=
(
x
1
+
x
2
,
y
1
y
2
)
(x_1,y_1)+(x_2,y_2)=(x_1+x_2,y_1y_2)
(
x
1
,
y
1
)
+
(
x
2
,
y
2
)
=
(
x
1
+
x
2
,
y
1
y
2
)
k
⋅
(
x
,
y
)
=
(
k
x
,
y
k
)
k\cdot(x,y)=(kx,y^k)
k
⋅
(
x
,
y
)
=
(
k
x
,
y
k
)
19.4F_Mid_Builder_$\tkcth{8.6.}\tkco{x}$_$\key{original \#1.}\tkct{d}$
The vectors
[
−
7
1
0
0
0
]
\begin{bmatrix} -7 \\ 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}
−
7
1
0
0
0
,
[
−
1
0
3
1
0
]
\begin{bmatrix} -1 \\ 0 \\ 3 \\ 1 \\ 0 \end{bmatrix}
−
1
0
3
1
0
,
[
2
0
−
5
0
1
]
\begin{bmatrix} 2 \\ 0 \\ -5 \\ 0 \\ 1 \end{bmatrix}
2
0
−
5
0
1
are a basis for some subspace of
R
5
\mathbb{R}^5
R
5
.
Show that
U
=
{
(
a
b
c
d
)
∈
M
2
×
2
(
R
)
∣
a
+
d
=
b
+
c
}
U=\Big\{ \begin{pmatrix} a&b\\ c&d \end{pmatrix}\in M_{2\times 2}(\mathbb{R})\;|\; a+d=b+c\Big\}
U
=
{
(
a
c
b
d
)
∈
M
2
×
2
(
R
)
∣
a
+
d
=
b
+
c
}
is a subspace of
M
2
×
2
(
R
)
M_{2\times2}(\mathbb{R})
M
2
×
2
(
R
)
over
R
\mathbb{R}
R
. Find a basis for
U
U
U
and find its dimension.
Show that
W
=
{
p
(
x
)
∈
P
2
(
R
)
∣
p
(
−
1
)
=
0
}
W=\Big\{ p(x)\in P_2(\mathbb{R})\;|\; p(-1)=0\Big\}
W
=
{
p
(
x
)
∈
P
2
(
R
)
∣
p
(
−
1
)
=
0
}
is a subspace of
P
2
(
R
)
P_2(\mathbb{R})
P
2
(
R
)
over
R
\mathbb{R}
R
. Find a basis for
W
W
W
and find its dimension.
Dimension of a subspace
In the vector space
P
2
P_2
P
2
, what is the dimension of the subspace that contains all polynomials of the form
p
(
x
)
=
a
+
b
x
+
c
x
2
p(x)=a+bx+cx^2
p
(
x
)
=
a
+
b
x
+
c
x
2
, such that
a
=
b
+
c
a=b+c
a
=
b
+
c
?
Practice Question: Dimension
For
subspaces
U
,
V
U,V
U
,
V
of
R
n
\mathbb{R}^n
R
n
, prove that
d
i
m
(
U
+
V
)
=
d
i
m
(
U
)
+
d
i
m
(
V
)
−
d
i
m
(
U
∩
V
)
dim(U+V) = dim(U)+dim(V)-dim(U\cap V)
d
im
(
U
+
V
)
=
d
im
(
U
)
+
d
im
(
V
)
−
d
im
(
U
∩
V
)
Bases of Subspaces
Suppose that
U
,
V
U,V
U
,
V
are subspaces of
R
4
\mathbb{R}^4
R
4
, with
d
i
m
(
U
)
=
2
,
d
i
m
(
V
)
=
3
dim(U)=2,dim(V)=3
d
im
(
U
)
=
2
,
d
im
(
V
)
=
3
. Show that there is a nonzero vector in
R
4
\mathbb{R}^4
R
4
which is in both
U
U
U
and
V
V
V
.
Basis of a Subspace
Find a basis for the following subspace of
M
2
×
2
M_{2\times2}
M
2
×
2
and state its dimension
Q
=
{
[
a
b
c
d
]
∈
M
2
×
2
∣
a
+
b
=
0
,
c
=
2
d
}
Q=\left\{\left[\begin{array}{rr} a&b\\ c&d \end{array}\right]\in M_{2\times2}\;\Big|\; a+b=0, c=2d\right\}
Q
=
{
[
a
c
b
d
]
∈
M
2
×
2
a
+
b
=
0
,
c
=
2
d
}
Basis of a Subspace
Find a basis for the following subspace of
R
3
\mathbb{R}^3
R
3
and state its dimension
S
=
{
x
⃗
∈
R
3
∣
[
1
−
2
−
1
]
⋅
x
⃗
=
0
⃗
}
S=\left\{\vec x\in\mathbb{R}^3\;\Big|\;\ \left[\begin{array}{rrr} 1&-2&-1\\ \end{array}\right]\cdot\vec x=\vec 0\right\}
S
=
{
x
∈
R
3
[
1
−
2
−
1
]
⋅
x
=
0
}
Practice Question: Subspaces
Which of the following subsets are subspaces?
U
=
{
(
x
,
y
,
z
)
∈
R
3
∣
x
=
2
y
,
z
=
1
}
U=\{(x,y,z)\in \mathbb{R}^3\;|\;x=2y,z=1\}
U
=
{(
x
,
y
,
z
)
∈
R
3
∣
x
=
2
y
,
z
=
1
}
W
=
{
w
⃗
∈
R
4
∣
A
w
⃗
=
5
w
⃗
}
W=\{\vec w \in \mathbb{R}^4\;|\;A\vec w = 5 \vec w\}
W
=
{
w
∈
R
4
∣
A
w
=
5
w
}
where
A
A
A
is a
4
×
4
4\times 4
4
×
4
matrix
Practice Question: Subspaces
For the following subset
U
U
U
of
R
3
\mathbb{R}^3
R
3
, select the statements which are true:
U
=
{
(
x
,
y
,
z
)
∈
R
3
∣
x
=
2
y
,
z
=
1
}
U=\{(x,y,z)\in \mathbb{R}^3\;|\;x=2y,z=1\}
U
=
{(
x
,
y
,
z
)
∈
R
3
∣
x
=
2
y
,
z
=
1
}
Practice Question: Subspaces
For the following subset
W
W
W
of
R
3
\mathbb{R}^3
R
3
, select the statements which are true:
W
=
{
(
a
,
b
,
c
)
∈
R
3
∣
c
<
0
}
W=\{(a,b,c)\in \mathbb{R}^3\;|\;c<0\}
W
=
{(
a
,
b
,
c
)
∈
R
3
∣
c
<
0
}
Give an example of a non-empty subset of
R
2
\mathbb{R}^2
R
2
that is closed under scalar multiplication, but is not a subspace.
Let
V
V
V
be a real vector space. Given two subspaces,
U
U
U
and
W
W
W
, is their union ever also a subspace? If so, what is the condition required for their union to be a subspace?