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Linear Independence and Span concepts
Related Topics
Wize University Linear Algebra Textbook > Vector Spaces and Subspaces
Span
4 Activities
Wize University Linear Algebra Textbook > Vector Spaces and Subspaces
Linear Independence
4 Activities
Which of the following statements are true about the set of vectors
{
(
1
,
1
,
−
1
,
0
)
,
(
−
1
,
3
,
−
1
,
1
)
,
(
2
,
0
,
−
3
,
4
)
}
\{(1,1,-1,0),(-1,3,-1,1),(2,0,-3,4)\}
{(
1
,
1
,
−
1
,
0
)
,
(
−
1
,
3
,
−
1
,
1
)
,
(
2
,
0
,
−
3
,
4
)}
?
The set spans
R
4
\mathbb{R}^4
R
4
and is independent
The set does not span
R
4
\mathbb{R}^4
R
4
and is independent
The set spans
R
4
\mathbb{R}^4
R
4
and is not independent
The set does not span
R
4
\mathbb{R}^4
R
4
and is not independent
None of the above
I don't know
Check Submission
More Span Questions:
Practice: Span
Show that for all
w
⃗
,
x
⃗
,
y
⃗
,
z
⃗
∈
R
n
\vec w,\vec x,\vec y, \vec z\in\mathbb{R}^n
w
,
x
,
y
,
z
∈
R
n
, if
w
⃗
∈
S
p
a
n
{
x
⃗
,
y
⃗
,
z
⃗
}
\vec w\in Span\{\vec x, \vec y, \vec z\}
w
∈
S
p
an
{
x
,
y
,
z
}
then
w
⃗
∈
S
p
a
n
{
x
⃗
−
y
⃗
,
y
⃗
−
z
⃗
,
z
⃗
}
\vec w\in Span\{\vec x-\vec y, \vec y-\vec z,\vec z\}
w
∈
S
p
an
{
x
−
y
,
y
−
z
,
z
}
.
When is a Vector in the Span of Others
Determine if
x
⃗
=
(
16
,
−
15
,
8
)
∈
Span
{
u
⃗
,
v
⃗
,
w
⃗
}
\vec x=(16,-15, 8) \in \text{Span}\{\vec u, \vec v, \vec w\}
x
=
(
16
,
−
15
,
8
)
∈
Span
{
u
,
v
,
w
}
if:
u
⃗
=
(
1
,
1
,
1
)
,
v
⃗
=
(
1
,
1
,
0
)
,
w
⃗
=
(
1
,
0
,
−
1
)
\vec u = (1,1,1), \vec v = (1,1,0), \vec w =(1,0,-1)
u
=
(
1
,
1
,
1
)
,
v
=
(
1
,
1
,
0
)
,
w
=
(
1
,
0
,
−
1
)
Span
Practice: Span
Let
u
⃗
=
⟨
1
,
0
,
1
⟩
,
v
⃗
=
⟨
1
,
1
,
1
⟩
,
w
⃗
=
⟨
1
,
−
1
,
1
⟩
\vec u = \lang 1,0,1\rang,\ \vec v = \lang 1,1,1\rang,\ \vec w = \lang 1,-1,1 \rang
u
=
⟨
1
,
0
,
1
⟩
,
v
=
⟨
1
,
1
,
1
⟩
,
w
=
⟨
1
,
−
1
,
1
⟩
be vectors in
R
3
\reals^3
R
3
, and suppose
b
⃗
=
⟨
3
,
y
,
z
⟩
∈
span
(
{
u
⃗
,
v
⃗
,
w
⃗
}
)
\vec b = \lang 3,y,z \rang \in \text{span}(\{\vec u, \vec v, \vec w\})
b
=
⟨
3
,
y
,
z
⟩
∈
span
({
u
,
v
,
w
})
.
Find the value of
z
z
z
.
Practice: Span
Let
u
⃗
=
(
2
,
2
,
1
)
,
v
⃗
=
(
2
,
1
,
2
)
,
w
⃗
=
(
2
,
5
,
−
2
)
,
x
⃗
=
(
x
,
y
,
z
)
\vec u = (2,2,1), \vec v = (2,1,2), \vec w = (2,5,-2), \vec x = (x,y,z)
u
=
(
2
,
2
,
1
)
,
v
=
(
2
,
1
,
2
)
,
w
=
(
2
,
5
,
−
2
)
,
x
=
(
x
,
y
,
z
)
be vectors in
R
3
\mathbb{R}^3
R
3
Find the conditions that the values
x
,
y
,
z
x,y,z
x
,
y
,
z
must satisfy in order for
x
⃗
\vec x
x
to be in
Span
(
{
u
⃗
,
v
⃗
,
w
⃗
}
)
\text{Span}(\{\vec u, \vec v, \vec w\})
Span
({
u
,
v
,
w
})
Span
Practice: Span
Let
u
⃗
=
⟨
1
,
0
,
1
⟩
,
v
⃗
=
⟨
1
,
1
,
1
⟩
,
w
⃗
=
⟨
1
,
−
1
,
1
⟩
\vec u = \lang 1,0,1\rang,\ \vec v = \lang 1,1,1\rang,\ \vec w = \lang 1,-1,1 \rang
u
=
⟨
1
,
0
,
1
⟩
,
v
=
⟨
1
,
1
,
1
⟩
,
w
=
⟨
1
,
−
1
,
1
⟩
be vectors in
R
3
\reals^3
R
3
, and suppose
b
⃗
=
⟨
3
,
y
,
z
⟩
∈
span
(
{
u
⃗
,
v
⃗
,
w
⃗
}
)
\vec b = \lang 3,y,z \rang \in \text{span}(\{\vec u, \vec v, \vec w\})
b
=
⟨
3
,
y
,
z
⟩
∈
span
({
u
,
v
,
w
})
.
Find the value of
z
z
z
.
Span
Select all of the vectors that are in
Span
(
u
⃗
,
v
⃗
)
\text{Span}(\vec u, \vec v)
Span
(
u
,
v
)
where
u
⃗
=
(
−
1
,
1
,
2
)
,
v
⃗
=
(
2
,
1
,
−
3
)
\vec u=(-1,1,2), \vec v=(2,1,-3)
u
=
(
−
1
,
1
,
2
)
,
v
=
(
2
,
1
,
−
3
)
133 - FML 3 - 18.1W - e.g. 31
Knowing that
[
0
6
3
3
0
−
7
1
5
1
−
1
1
3
]
→
RREF
[
1
0
0
0
1
0
0
0
1
0
0
0
]
\bcb{\begin{bmatrix}%[c] 0 & 6 & 3 \\ 3 & 0 & -7 \\ 1 & 5 & 1 \\ -1 & 1 & 3 \end{bmatrix} % \xrightarrow{\text{{\tiny RREF}}} % \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}}
0
3
1
−
1
6
0
5
1
3
−
7
1
3
RREF
1
0
0
0
0
1
0
0
0
0
1
0
,
determine whether the vector
v
⃗
1
=
<
0
,
3
,
1
,
−
1
>
\bcb{\vec{v}_1 = \big< 0,3,1,-1\big>}
v
1
=
⟨
0
,
3
,
1
,
−
1
⟩
is in the span of the vectors
v
⃗
2
=
<
6
,
0
,
5
,
1
>
\bcb{\vec{v}_2 = \big< 6,0,5,1\big>}
v
2
=
⟨
6
,
0
,
5
,
1
⟩
and
v
⃗
3
=
<
3
,
−
7
,
1
,
3
>
\bcb{\vec{v}_3 = \big< 3,-7,1,3\big>}
v
3
=
⟨
3
,
−
7
,
1
,
3
⟩
.
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
.
19.4F_Final_Builder_Ch_4.6_Span_Line_From_FAQ_$\tkco{eg}$_$\key{Final}$_Builder_$\tkcth{4.6.}\tkcf{6}$_$\key{|Strc>5.2}$
Describe the set of all linear combinations (sums of constant multiples of) the vectors:
v
⃗
=
[
1
1
1
]
\vv = \colvecth{1}{1}{1}
v
=
1
1
1
and
w
⃗
=
[
−
1
2
1
]
\vw = \colvecth{-1}{2}{1}
w
=
−
1
2
1
.
133 - FML 3 - 18.1W - e.g. 30
Given the standard basis vectors for
R
2
\bcb{\mathbb{R}^{2}}
R
2
,
viz.
ı
^
\bcb{\ihat}
^
and
ȷ
^
\bcb{\jhat}
^
and the vector
u
⃗
=
[
0
2
]
\bcb{\vec{u} = \begin{bmatrix} 0 \\ 2 \end{bmatrix}}
u
=
[
0
2
]
, determine whether
ı
^
\bcb{\ihat}
^
is in the
span
{
u
⃗
,
ȷ
^
}
\bcb{ \vspan{\vec{u}, \jhat} }
span
{
u
,
^
}
.
Briefly justify your answer.
Let
(
V
,
+
,
⋅
)
(V, +, \cdot)
(
V
,
+
,
⋅
)
be a real vector space. If
v
1
,
…
,
v
n
v_1, \dots, v_n
v
1
,
…
,
v
n
is a set of vectors that spans the entirety of
V
V
V
, prove that the following set of vectors also spans
V
V
V
.
v
1
−
v
2
,
v
2
−
v
3
,
…
,
v
n
−
1
−
v
n
,
v
n
v_1 - v_2, v_2 - v_3, \dots, v_{n - 1} - v_n, v_n
v
1
−
v
2
,
v
2
−
v
3
,
…
,
v
n
−
1
−
v
n
,
v
n
19.4F_Final_Builder_Ch_4.6_Span_Line_From_FAQ_$\tkco{eg}$_$\key{Final}$_Builder_$\tkcth{4.6.}\tkcf{4}$_$\key{|Strc>5.2}$
Describe the set of all constant multiples of the vector:
v
⃗
=
[
−
1
2
1
]
\vv = \colvecth{-1}{2}{1}
v
=
−
1
2
1
.
19.4F_Final_Builder_Ch_4.6_Span_Line_From_FAQ_$\tkco{eg}$_$\key{Final}$_Builder_$\tkcth{4.6.}\tkcf{2}$_$\key{|Strc>5.2}$
Describe the set of all constant multiples of the vector
v
⃗
=
[
1
1
]
\vv = \colvec{1}{1}
v
=
[
1
1
]
.
Practice Question: Arbitrary Vector as Linear Combination
Show that every vector in
R
2
\mathbb{R}^2
R
2
is a linear combination of
(
1
,
−
1
)
(1,-1)
(
1
,
−
1
)
and
(
2
,
1
)
(2,1)
(
2
,
1
)
.
Arbitrary Vector as Linear Combination
Show every vector in
R
3
\mathbb{R}^3
R
3
is a combination of
(
8
,
0
,
0
)
,
(
0
,
1
,
1
)
,
(
0
,
2
,
3
)
(8,0,0),(0,1,1),(0,2,3)
(
8
,
0
,
0
)
,
(
0
,
1
,
1
)
,
(
0
,
2
,
3
)
.
Suppose that
S
=
s
p
a
n
(
[
1
−
1
2
]
,
[
−
3
3
−
6
]
,
[
1
2
1
]
,
[
5
4
7
]
)
S=span\left(\begin{bmatrix}1\\-1\\2\end{bmatrix},\begin{bmatrix}-3\\3\\-6\end{bmatrix},\begin{bmatrix}1\\2\\1\end{bmatrix},\begin{bmatrix}5\\4\\7\end{bmatrix}\right)
S
=
s
p
an
1
−
1
2
,
−
3
3
−
6
,
1
2
1
,
5
4
7
, which of the following statements is true?
More Linear Independence Questions:
Linearly independent vectors
Determine variable x such that 𝒖 = (1,1,1) ,𝒗 = (3,2,1) and 𝒘 = (𝑥, 0,1) are linearly dependent.
Linearly independent vectors
Determine variable x such that 𝒖 = (1,1,1) ,𝒗 = (3,2,1) and 𝒘 = (𝑥, 0,1) are linearly dependent.
Determine r and s such that (1,-1,4,1), (-1,1,0,0) and (2,r,s,3) are linearly dependent.
Linearly independent vectors
Is this set of vectors {[1 3], [2,-2]} linearly independent? Write [4,-2] as a linear combination of this set.
Linear Independence
Example: Linear Independence
Let
u
⃗
=
[
4
0
−
2
]
\vec u = \begin{bmatrix} 4\\ 0\\ -2\\ \end{bmatrix}
u
=
4
0
−
2
,
v
⃗
=
[
1
3
7
]
\vec v= \begin{bmatrix} 1\\ 3\\ 7\\ \end{bmatrix}
v
=
1
3
7
, and
w
⃗
=
[
3
3
5
]
\vec w = \begin{bmatrix} 3\\ 3\\ 5\\ \end{bmatrix}
w
=
3
3
5
.
Determine if the set
{
u
⃗
,
v
⃗
,
w
⃗
}
\{\vec u, \vec v, \vec w\}
{
u
,
v
,
w
}
is linearly independent.
Practice Question: Linear Independence
Use the definition of linear independence to show that the vectors
(
1
,
0
,
3
)
,
(
0
,
−
1
,
0
)
,
(
2
,
2
,
2
)
∈
R
3
(1,0,3),(0,-1,0),(2,2,2) \in\mathbb{R}^3
(
1
,
0
,
3
)
,
(
0
,
−
1
,
0
)
,
(
2
,
2
,
2
)
∈
R
3
are linearly independent.
Practice: Linear Independence
Show that if
x
⃗
+
z
⃗
,
x
⃗
+
3
⃗
z
,
y
⃗
+
z
⃗
\vec x+\vec z, \vec x+\vec 3z, \vec y+\vec z
x
+
z
,
x
+
3
z
,
y
+
z
are linearly independent vectors in
R
n
\mathbb{R}^n
R
n
then
x
⃗
,
y
⃗
,
z
⃗
\vec x,\vec y,\vec z
x
,
y
,
z
are also linearly independent vectors in
R
n
\mathbb{R}^n
R
n
.
Practice Question: Linear Independence
Use the definition of linear independence to show that the vectors
(
1
,
0
,
3
)
,
(
0
,
−
1
,
0
)
,
(
2
,
2
,
2
)
∈
R
3
(1,0,3),(0,-1,0),(2,2,2) \in\mathbb{R}^3
(
1
,
0
,
3
)
,
(
0
,
−
1
,
0
)
,
(
2
,
2
,
2
)
∈
R
3
are linearly independent
Linear Independence
Practice: Linear Independence
Let
{
v
⃗
1
,
v
⃗
2
,
v
⃗
3
}
\{\vec v_1,\vec v_2,\vec v_3\}
{
v
1
,
v
2
,
v
3
}
be a linearly independent set of vectors in a vector space
V
V
V
.
Define the following vectors in
V
V
V
, where
k
∈
R
k \in \reals
k
∈
R
:
Practice: Linear Independence
Let
u
⃗
=
(
−
1
,
1
,
0
,
1
)
\vec u = (-1,1,0,1)
u
=
(
−
1
,
1
,
0
,
1
)
,
v
⃗
=
(
1
,
0
,
−
1
,
1
)
\vec v = (1,0,-1,1)
v
=
(
1
,
0
,
−
1
,
1
)
, and
w
⃗
=
(
0
,
−
1
,
1
,
−
2
)
\vec w=(0,-1,1,-2)
w
=
(
0
,
−
1
,
1
,
−
2
)
Determine if the set
{
u
⃗
,
v
⃗
,
w
⃗
}
\{\vec u, \vec v, \vec w\}
{
u
,
v
,
w
}
is linearly independent
Practice: Linear Independence
Let
{
v
⃗
1
,
v
⃗
2
,
v
⃗
3
}
\{\vec v_1,\vec v_2,\vec v_3\}
{
v
1
,
v
2
,
v
3
}
be a linearly independent set of vectors in a vector space
V
V
V
Now define the following vectors in
V
V
V
:
w
⃗
1
=
2
v
⃗
2
+
k
v
⃗
3
w
⃗
2
=
v
⃗
1
+
k
v
⃗
2
w
⃗
3
=
k
v
⃗
1
−
4
v
⃗
3
\vec w_1=2\vec v_2+k\vec v_3\qquad \vec w_2=\vec v_1+k\vec v_2 \qquad \vec w_3=k\vec v_1-4\vec v_3
w
1
=
2
v
2
+
k
v
3
w
2
=
v
1
+
k
v
2
w
3
=
k
v
1
−
4
v
3
Linear Independence
Practice: Linear Independence
Let
{
v
⃗
1
,
v
⃗
2
,
v
⃗
3
}
\{\vec v_1,\vec v_2,\vec v_3\}
{
v
1
,
v
2
,
v
3
}
be a linearly independent set of vectors in a vector space
V
V
V
.
Define the following vectors in
V
V
V
, where
k
∈
R
k \in \reals
k
∈
R
:
Determine which ones are linearly independent:
Linear Independence
Example: Linear Independence
Let
u
⃗
=
(
1
,
2
,
1
)
\vec u = (1,2,1)
u
=
(
1
,
2
,
1
)
,
v
⃗
=
(
1
,
3
,
2
)
\vec v=(1,3,2)
v
=
(
1
,
3
,
2
)
, and
w
⃗
=
(
7
,
17
,
a
2
+
1
)
\vec w = (7,17,a^2+1)
w
=
(
7
,
17
,
a
2
+
1
)
.
Determine the value(s) of
a
a
a
that will make the set
{
u
⃗
,
v
⃗
,
w
⃗
}
\{\vec u, \vec v, \vec w\}
{
u
,
v
,
w
}
linearly independent.
Linear Independence
Let
u
⃗
=
[
4
0
−
2
]
\vec u = \begin{bmatrix} 4\\ 0\\ -2\\ \end{bmatrix}
u
=
4
0
−
2
,
v
⃗
=
[
1
3
7
]
\vec v= \begin{bmatrix} 1\\ 3\\ 7\\ \end{bmatrix}
v
=
1
3
7
, and
w
⃗
=
[
3
3
5
]
\vec w = \begin{bmatrix} 3\\ 3\\ 5\\ \end{bmatrix}
w
=
3
3
5
.
Determine if the set
{
u
⃗
,
v
⃗
,
w
⃗
}
\{\vec u, \vec v, \vec w\}
{
u
,
v
,
w
}
is linearly independent.
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
.
133 - FML 3 - 18.1W - e.g. 32
Show that the vectors
v
⃗
1
=
(
1
,
2
,
3
,
4
)
\bcb{\vec{v}_1 = (1,2,3,4)}
v
1
=
(
1
,
2
,
3
,
4
)
,
v
⃗
2
=
(
0
,
1
,
0
,
−
1
)
\bcb{\vec{v}_2 = (0,1,0,-1)}
v
2
=
(
0
,
1
,
0
,
−
1
)
, and
v
⃗
3
=
(
1
,
3
,
3
,
3
)
\bcb{\vec{v}_3 = (1,3,3,3)}
v
3
=
(
1
,
3
,
3
,
3
)
form a linearly dependent set, then express
each
vector as a linear combination of the other two.
v
⃗
1
=
\vv_1 =
v
1
=
-1
v
⃗
2
\vv_2
v
2
+
1
v
⃗
3
\vv_3
v
3
v
⃗
2
=
\vv_2 =
v
2
=
-1
v
⃗
1
\vv_1
v
1
+
1
v
⃗
3
\vv_3
v
3
Let our real vector space be
(
R
3
,
+
,
⋅
)
(\mathbb{R}^3, +, \cdot)
(
R
3
,
+
,
⋅
)
. Are the following set of vectors linearly independent? If not, write one of the vectors as as linear combination of the others.
(
−
1
3
4
)
,
(
3
1
−
2
)
,
(
13
7
5
)
,
(
1
2
1
)
\left ( \begin{array}{c} -1\\3\\4 \end{array} \right ), \left ( \begin{array}{c} 3\\1\\-2 \end{array} \right ), \left ( \begin{array}{c} 13\\7\\5 \end{array} \right ), \left ( \begin{array}{c} 1\\2\\1 \end{array} \right )
−
1
3
4
,
3
1
−
2
,
13
7
5
,
1
2
1
Let our real vector space be
(
R
3
,
+
,
⋅
)
(\mathbb{R}^3, +, \cdot)
(
R
3
,
+
,
⋅
)
. Are the following set of vectors linearly independent? If not, write one of the vectors as a linear combination of the others.
(
1
2
3
)
,
(
0
1
1
)
,
(
−
1
1
1
)
\left ( \begin{array}{c} 1\\2\\3 \end{array} \right ), \left ( \begin{array}{c} 0\\1\\1 \end{array} \right ), \left ( \begin{array}{c} -1\\1\\1 \end{array} \right )
1
2
3
,
0
1
1
,
−
1
1
1
133- 05.4F - Final - PartI Question#7
For what values of α are the vectors
(
α
,
1
,
1
)
\bcb{(\alpha, 1, 1)}
(
α
,
1
,
1
)
,
(
1
,
α
,
α
)
\bcb{(1, \alpha, \alpha)}
(
1
,
α
,
α
)
and
(
8
,
7
,
α
)
\bcb{(8, 7, \alpha) }
(
8
,
7
,
α
)
linearly dependent ?
133- 05.4F - Final - PartII Question#3b
Suppose that the vectors
{
v
1
,
v
2
,
v
3
}
\bcb{ \{v1, v2, v3\} }
{
v
1
,
v
2
,
v
3
}
are linearly independent.
If
a
1
=
4
v
1
+
2
v
2
−
2
v
3
\bcb{a1 = 4v1 + 2v2 − 2v3}
a
1
=
4
v
1
+
2
v
2
−
2
v
3
,
a
2
=
v
1
+
3
v
2
−
v
3
\bcb{a2 = v1 + 3v2 − v3}
a
2
=
v
1
+
3
v
2
−
v
3
, and
a
3
=
3
v
1
−
6
v
2
\bcb{a3 = 3v1 − 6v2}
a
3
=
3
v
1
−
6
v
2
, show that the vectors {a1, a2, a3} are not linearly independent.
133- 05.4F - Final - PartII Question#3a
Suppose that the vectors
{
v
1
,
v
2
,
v
3
}
\bcb{ \{v1, v2, v3\} }
{
v
1
,
v
2
,
v
3
}
are linearly independent.
If
w
1
=
v
1
−
v
2
+
v
3
\bcb{w1 = v1 − v2 + v3}
w
1
=
v
1
−
v
2
+
v
3
,
w
2
=
v
1
−
v
3
\bcb{w2 = v1 − v3}
w
2
=
v
1
−
v
3
, and
w
3
=
v
1
+
v
2
+
v
3
\bcb{w3 = v1 + v2 + v3}
w
3
=
v
1
+
v
2
+
v
3
, show that the vectors
{
w
1
,
w
2
,
w
3
}
\bcb{ \{w1, w2, w3\}}
{
w
1
,
w
2
,
w
3
}
are linearly independent.
Determine r and s such that (1,-1,4,1), (-1,1,0,0) and (2,r,s,3) are linearly dependent.
Given that
u
⃗
\vec{u}
u
and
v
⃗
\vec{v}
v
are linearly independent, which of the following sets is also linearly independent?