Use Exercise 37 to obtain necessary and sufficient conditions for a set { u , v } of two vectors to be linearly dependent. Determine by inspection whether each of the following sets is linearly dependent or linearly independent. a) { 1 + x , x 2 } . b) { x , e x } . c) { x , 3 x } . d) { [ − 1 2 1 3 ] , [ 2 − 4 − 2 − 6 ] } . e) { [ 0 0 0 0 ] , [ 1 0 0 1 ] } 3 7 . Let S = { v 1 , v 2 , ... , v n } be a subset of a vector space V , where n ≥ 2 . Prove that set S is linearly dependent if and only if at least one of the vectors, v j , can be expressed as a linear combination of the remaining vectors.
Use Exercise 37 to obtain necessary and sufficient conditions for a set { u , v } of two vectors to be linearly dependent. Determine by inspection whether each of the following sets is linearly dependent or linearly independent. a) { 1 + x , x 2 } . b) { x , e x } . c) { x , 3 x } . d) { [ − 1 2 1 3 ] , [ 2 − 4 − 2 − 6 ] } . e) { [ 0 0 0 0 ] , [ 1 0 0 1 ] } 3 7 . Let S = { v 1 , v 2 , ... , v n } be a subset of a vector space V , where n ≥ 2 . Prove that set S is linearly dependent if and only if at least one of the vectors, v j , can be expressed as a linear combination of the remaining vectors.
Solution Summary: The author explains that a set is linearly dependent if at least one of the vectors, v_j, can be expressed.
Use Exercise
37
to obtain necessary and sufficient conditions for a set
{
u
,
v
}
of two vectors to be linearly dependent. Determine by inspection whether each of the following sets is linearly dependent or linearly independent.
a)
{
1
+
x
,
x
2
}
.
b)
{
x
,
e
x
}
.
c)
{
x
,
3
x
}
.
d)
{
[
−
1
2
1
3
]
,
[
2
−
4
−
2
−
6
]
}
.
e)
{
[
0
0
0
0
]
,
[
1
0
0
1
]
}
3
7
. Let
S
=
{
v
1
,
v
2
,
...
,
v
n
}
be a subset of a vector space
V
, where
n
≥
2
. Prove that set
S
is linearly dependent if and only if at least one of the vectors,
v
j
, can be expressed as a linear combination of the remaining vectors.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
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