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133- 05.4F - Final - PartI Question#7
Related Topics
Wize University Linear Algebra Textbook > Vector Spaces and Subspaces
Linear Independence
4 Activities
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 ?
0, 1, 7
-7, 1, -1
7, -7, 1
7,- 7 , -1
7, 1, -1
I don't know
Check Submission
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
:
Linear Independence and Span concepts
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
)}
?
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 - 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?