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$\tkct{Mock F1}$ 19.4F_Mid_Builder_$\tkcth{8.6.}\tkco{x}$_$\key{original \#1.}\…
Related Topics
Wize University Linear Algebra Textbook > Vector Spaces and Subspaces
Basis and Dimension
4 Activities
If vectors
u
⃗
\vu
u
,
v
⃗
\vv
v
,
x
⃗
\vx
x
and
y
⃗
\vec{y}
y
are vectors in
R
7
\mathbb{R}^7
R
7
such that
u
⃗
=
−
2
v
⃗
+
0
x
⃗
+
4
y
⃗
\vu = -2\vv + 0\vx +4\vec{y}
u
=
−
2
v
+
0
x
+
4
y
, then a basis for
span
{
u
⃗
,
v
⃗
,
x
⃗
,
y
⃗
}
\text{span}\big\{ \vu,\, \vv,\, \vx,\, \vec{y} \big\}
span
{
u
,
v
,
x
,
y
}
is
{
u
⃗
,
v
⃗
,
y
⃗
}
\big\{ \vu,\, \vv,\, \vec{y} \big\}
{
u
,
v
,
y
}
.
Answer
I don't know
Check Submission
More Basis and Dimension Questions:
Example: Linear Independence (pre-Matrix)
Show that for
x
⃗
,
y
⃗
∈
R
2
\vec x,\vec y\in \mathbb{R}^2
x
,
y
∈
R
2
, if
S
p
a
n
{
x
⃗
,
y
⃗
}
=
R
2
Span\{\vec x,\vec y\}=\mathbb{R}^2
S
p
an
{
x
,
y
}
=
R
2
, then
x
⃗
,
y
⃗
\vec x,\vec y
x
,
y
are
linearly independent
.
Practice Question: Bases
Practice Question: Bases
For each of the following
subspaces
, find the
dimension
of the
subspace
by giving a specific example of a
basis
:
Q
:
=
s
p
a
n
{
(
0
,
0
,
0
,
0
)
,
(
1
,
1
,
0
,
0
)
,
(
0
,
0
,
1
,
1
)
,
(
1
,
1
,
1
,
1
)
}
⊆
R
4
Q:=span\{(0,0,0,0), (1,1,0,0),(0,0,1,1),(1,1,1,1)\}\subseteq\mathbb{R}^4
Q
:=
s
p
an
{(
0
,
0
,
0
,
0
)
,
(
1
,
1
,
0
,
0
)
,
(
0
,
0
,
1
,
1
)
,
(
1
,
1
,
1
,
1
)}
⊆
R
4
Linear Independence and Basis
Let
v
⃗
1
,
v
⃗
2
,
v
⃗
3
,
v
⃗
4
\vec v_1,\vec v_2,\vec v_3, \vec v_4
v
1
,
v
2
,
v
3
,
v
4
be any four vectors in
R
4
\mathbb{R}^4
R
4
Explain why the set
C
=
{
v
⃗
1
−
v
⃗
2
,
v
⃗
2
−
v
⃗
3
,
v
⃗
3
−
v
⃗
4
,
v
⃗
4
−
v
⃗
1
}
C=\{\vec v_1-\vec v_2,\vec v_2-\vec v_3,\vec v_3-\vec v_4,\vec v_4-\vec v_1\}
C
=
{
v
1
−
v
2
,
v
2
−
v
3
,
v
3
−
v
4
,
v
4
−
v
1
}
cannot be a basis for
R
4
\mathbb{R}^4
R
4
133-WML2_Quiz-19.4F_eg_17
It is known that the set
{
v
⃗
1
,
v
⃗
2
,
…
,
v
⃗
n
}
\{\vv_1,\, \vv_2,\, \dots \,,\, \vv_n\}
{
v
1
,
v
2
,
…
,
v
n
}
is a basis for
R
n
\mathbb{R^n}
R
n
. If
w
⃗
∈
\vw \in
w
∈
R
n
\mathbb{R^n}
R
n
, is the set
{
v
⃗
1
,
v
⃗
2
,
…
,
v
⃗
n
,
w
⃗
}
\{\vv_1,\, \vv_2,\, \dots \,,\, \vv_n,\, \vw\}
{
v
1
,
v
2
,
…
,
v
n
,
w
}
linearly in/dependent? Briefly justify your answer (using mathematical notation is preferable to words, if possible).
Which of the following is a basis for
R
2
\mathbb{R}^2
R
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
)
Which of the following is a basis for
R
2
\mathbb{R}^2
R
2
?
Which of the following is a basis for
R
2
\mathbb{R}^2
R
2
?
Consider the vector space
M
2
,
2
(
R
)
\mathcal{M}_{2, 2} (\mathbb{R})
M
2
,
2
(
R
)
. What is the dimension of this vector space?
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
?
133 - FML 3 - 18.1W - e.g. 51
If it is known that the set of vectors
{
x
⃗
,
y
⃗
,
z
⃗
,
w
⃗
}
\bcb{\left\{ \vec{x},\, \vec{y},\, \vec{z},\, \vec{w} \right\}}
{
x
,
y
,
z
,
w
}
form a basis for
R
4
\bcb{\mathbb{R}^4}
R
4
,
show
that the set
{
x
⃗
+
k
z
⃗
,
y
⃗
,
z
⃗
,
w
⃗
}
\bcb{\left\{ \vec{x} + k\vec{z}, \vec{y},\, \vec{z},\, \vec{w} \right\}}
{
x
+
k
z
,
y
,
z
,
w
}
is also a basis, for any scalar
k
\bcb{k}
k
.
Verify if the following vectors form a basis for
R
3
\mathbb{R}^3
R
3
:
{
[
1
1
2
]
,
[
1
−
2
1
]
,
[
2
−
1
1
]
}
\left\{ \begin{bmatrix} 1 \\ 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 1 \\ -2 \\ 1 \end{bmatrix},\begin{bmatrix} 2 \\ -1 \\ 1 \end{bmatrix} \right\}
⎩
⎨
⎧
1
1
2
,
1
−
2
1
,
2
−
1
1
⎭
⎬
⎫
133 - FML 3 - 18.1W - e.g. 42
Are the vectors
[
1
3
1
0
]
,
[
2
−
1
0
0
]
,
[
0
2
1
−
1
]
,
[
3
6
−
3
−
2
]
\bcb{\begin{bmatrix} 1 \\ 3 \\ 1 \\ 0 \end{bmatrix}, \, \begin{bmatrix} 2 \\ -1 \\ 0 \\ 0 \end{bmatrix}, \, \begin{bmatrix} 0 \\ 2 \\ 1 \\ -1 \end{bmatrix} , \, \begin{bmatrix} 3 \\ 6 \\ -3 \\ -2 \end{bmatrix} }
1
3
1
0
,
2
−
1
0
0
,
0
2
1
−
1
,
3
6
−
3
−
2
a basis for
R
4
\bcb{\mathbb{R}^4}
R
4
?
133 - FML 3 - 18.1W - e.g. 44
Find a basis for the set of vectors
{
[
1
0
−
1
]
,
[
2
−
2
0
]
,
[
0
−
2
2
]
,
[
3
−
4
1
]
}
\bcb{\left\{ \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix},\, \begin{bmatrix} 2 \\ -2 \\ 0 \end{bmatrix},\, \begin{bmatrix} 0 \\ -2 \\ 2 \end{bmatrix}, \, \begin{bmatrix} 3 \\ -4 \\ 1\end{bmatrix} \right\}}
⎩
⎨
⎧
1
0
−
1
,
2
−
2
0
,
0
−
2
2
,
3
−
4
1
⎭
⎬
⎫
, and determine the dimension of their span.
Practice: Coordinates
Practice: Coordinates
Let
B
1
=
{
[
−
1
2
0
]
,
[
3
−
5
2
]
,
[
4
−
7
3
]
}
B_1= \left\{ \begin{bmatrix} -1\\ 2 \\0\\ \end{bmatrix} , \begin{bmatrix} 3\\ -5\\ 2\\ \end{bmatrix} , \begin{bmatrix} 4\\ -7\\ 3\\ \end{bmatrix} \right\}
B
1
=
⎩
⎨
⎧
−
1
2
0
,
3
−
5
2
,
4
−
7
3
⎭
⎬
⎫
and
B
2
=
{
[
1
−
4
3
]
,
[
5
2
−
2
]
,
[
4
−
7
0
]
}
B_2= \left\{ \begin{bmatrix} 1\\ -4 \\3\\ \end{bmatrix} , \begin{bmatrix} 5\\ 2\\ -2\\ \end{bmatrix} , \begin{bmatrix} 4\\ -7\\ 0\\ \end{bmatrix} \right\}
B
2
=
⎩
⎨
⎧
1
−
4
3
,
5
2
−
2
,
4
−
7
0
⎭
⎬
⎫
be two bases of
R
3
\mathbb{R}^3
R
3
.
If
[
x
⃗
]
B
1
=
[
−
4
8
−
7
]
[\vec x]_{\small B_1} = \begin{bmatrix} -4\\ 8\\ -7\\ \end{bmatrix}
[
x
]
B
1
=
−
4
8
−
7
, find:
Which of the following are a basis for the vector space of polynomials with degree at most 3?
Let
{
e
1
,
…
,
e
n
}
\{ e_1, \dots, e_n \}
{
e
1
,
…
,
e
n
}
be a basis for an
n
n
n
- dimensional vector space
V
V
V
, and let
L
:
V
→
W
L : V\rightarrow W
L
:
V
→
W
be a linear map, where
W
W
W
is also a vector space. What conditions are necessary for
{
L
(
e
1
)
,
…
,
L
(
e
n
)
}
\{ L(e_1), \dots, L(e_n) \}
{
L
(
e
1
)
,
…
,
L
(
e
n
)}
to be a basis for
W
W
W
?
Example: Linear Independence (pre-Matrix)
Example:
Show that for
x
⃗
,
y
⃗
∈
R
2
\vec x,\vec y\in \mathbb{R}^2
x
,
y
∈
R
2
, if
S
p
a
n
{
x
⃗
,
y
⃗
}
=
R
2
Span\{\vec x,\vec y\}=\mathbb{R}^2
S
p
an
{
x
,
y
}
=
R
2
, then
x
⃗
,
y
⃗
\vec x,\vec y
x
,
y
are
linearly independent
.
Basis and Dimension
Practice: Finding a Basis
Consider the vector space
M
2
×
2
M_{2\times2}
M
2
×
2
with the standard component-wise operations of addition and scalar multiplication
The subset
D
=
{
[
a
b
c
d
]
∈
M
2
×
2
∣
a
+
d
=
b
+
c
}
D=\left\{ \left[\begin{array}{cc} a&b\\ c&d \end{array}\right]\in M_{2\times2}\;\Big|\;a+d=b+c \right\}
D
=
{
[
a
c
b
d
]
∈
M
2
×
2
a
+
d
=
b
+
c
}
is a subspace of
M
2
×
2
M_{2\times2}
M
2
×
2
Finding a basis and the dimension of a subspace
In the real vector space
R
3
\mathbb{R}^3
R
3
with the standard component-wise operations of vector addition and scalar multiplication, consider the subspace:
W
=
{
(
x
,
y
,
z
)
∈
R
3
∣
y
+
3
z
=
0
}
W=\{(x,y,z)\in\mathbb{R}^3\;|\;y+3z=0\}
W
=
{(
x
,
y
,
z
)
∈
R
3
∣
y
+
3
z
=
0
}
Find a basis for
W
W
W
, and use your answer to state the dimension of
W
W
W
Concept Clarifier
Determine whether the set
T
=
{
(
1
,
0
,
0
)
,
(
0
,
−
2
,
2
)
,
(
1
,
1
,
1
)
}
T=\{ (1,0,0),(0,-2,2), (1,1,1) \}
T
=
{(
1
,
0
,
0
)
,
(
0
,
−
2
,
2
)
,
(
1
,
1
,
1
)}
is a basis of
R
3
\mathbb{R}^3
R
3
.
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
|
x
⃗
⋅
v
⃗
=
0
for every
v
⃗
∈
s
p
a
n
{
(
1
,
−
1
,
1
)
,
(
1
,
1
,
0
)
}
}
S:=\{\vec x\in\mathbb{R}^3\text{ | }\vec x\cdot\vec v=0\text{ for every } \vec v\in span\{(1,-1,1),(1,1,0)\}\}
S
:=
{
x
∈
R
3
|
x
⋅
v
=
0
for every
v
∈
s
p
an
{(
1
,
−
1
,
1
)
,
(
1
,
1
,
0
)}}
Example: Abstract Basis Proof
Suppose the set
B
=
{
v
1
→
,
v
2
→
,
v
3
→
}
B=\{\overrightarrow{v_1},\overrightarrow{v_2},\overrightarrow{v_3}\}
B
=
{
v
1
,
v
2
,
v
3
}
is a basis for a vector space
V
V
V
Define the vectors:
u
1
→
=
v
1
→
−
v
2
→
u
2
→
=
v
2
→
−
v
3
→
u
3
→
=
−
v
3
→
\overrightarrow{u_1}=\overrightarrow{v_1}-\overrightarrow{v_2} \hspace{1cm}\overrightarrow{u_2}=\overrightarrow{v_2}-\overrightarrow{v_3}\hspace{1cm}\overrightarrow{u_3}=-\overrightarrow{v_3}
u
1
=
v
1
−
v
2
u
2
=
v
2
−
v
3
u
3
=
−
v
3
Basis and Dimension
Example: Verifying a Basis
Let
V
V
V
be the vector space
{
(
x
,
y
)
∣
x
,
y
∈
R
+
}
\{(x,y)\;|\;x,y\in\mathbb{R}^+\}
{(
x
,
y
)
∣
x
,
y
∈
R
+
}
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_1x_2,\ y_1y_2)
(
x
1
,
y
1
)
+
(
x
2
,
y
2
)
=
(
x
1
x
2
,
y
1
y
2
)
Find a basis for a given space
Consider the vector space
M
2
×
2
M_{2\times2}
M
2
×
2
with the standard component-wise operations of addition and scalar multiplication
The subset
D
=
{
[
a
b
c
d
]
∈
M
2
×
2
∣
a
+
d
=
b
+
c
}
D=\left\{ \left[\begin{array}{cc} a&b\\ c&d \end{array}\right]\in M_{2\times2}\;\Big|\;a+d=b+c \right\}
D
=
{
[
a
c
b
d
]
∈
M
2
×
2
a
+
d
=
b
+
c
}
is a subspace of
M
2
×
2
M_{2\times2}
M
2
×
2
Find a basis for
D
D
D
and the dimension of
D
D
D
Show that a set is a basis
Let
A
=
[
1
−
1
1
]
A=\left[\begin{array}{rrr} 1&-1&1\\ \end{array}\right]
A
=
[
1
−
1
1
]
and consider the subspace of
R
3
\mathbb{R}^3
R
3
:
N
=
{
x
→
=
(
x
,
y
,
z
)
∈
R
3
∣
A
.
x
→
=
0
→
}
N=\{\overrightarrow{x}=(x,y,z)\in\mathbb{R}^3\;|\;A.\overrightarrow{x}=\overrightarrow{0}\}
N
=
{
x
=
(
x
,
y
,
z
)
∈
R
3
∣
A
.
x
=
0
}
Show that
B
=
{
(
1
,
1
,
0
)
,
(
−
1
,
0
,
1
)
}
B=\{(1,1,0), (-1,0,1)\}
B
=
{(
1
,
1
,
0
)
,
(
−
1
,
0
,
1
)}
is a basis for
N
N
N
Consider the space of polynomials of degree 3 or lower,
P
3
\mathcal{P}_3
P
3
. Give an example of a linearly independent set of vectors whose span is the entirety of
P
3
\mathcal{P}_3
P
3
.
For which value(s) of c do
[
1
1
0
]
,
[
2
c
1
]
,
[
c
4
3
]
\begin{bmatrix}1\\1\\0\end{bmatrix}, \begin{bmatrix}2\\c\\1\end{bmatrix}, \begin{bmatrix}c\\4\\3\end{bmatrix}
1
1
0
,
2
c
1
,
c
4
3
form a basis for
R
3
\mathbb{R}^3
R
3
?
Which of the following sets are a basis for
R
3
\mathbb{R}^3
R
3
?