Determine which of the following sets of vectors is a basis for the solution space to the differential equation y ″ − 16 y = 0 : S 1 = { e 4 x } , S 2 = { e 2 x , e 4 x , e − 4 x } , S 3 = { e 4 x , e 2 x } , S 4 = { e 4 x , e − 4 x } , S 5 = { e 4 x , 7 e 4 x } , S 6 = { cosh 4 x , sinh 4 x } .
Determine which of the following sets of vectors is a basis for the solution space to the differential equation y ″ − 16 y = 0 : S 1 = { e 4 x } , S 2 = { e 2 x , e 4 x , e − 4 x } , S 3 = { e 4 x , e 2 x } , S 4 = { e 4 x , e − 4 x } , S 5 = { e 4 x , 7 e 4 x } , S 6 = { cosh 4 x , sinh 4 x } .
Solution Summary: The author explains that a vector space is linearly independent when its dimension is 2.
Determine which of the following sets of vectors is a basis for the solution space to the differential equation
y
″
−
16
y
=
0
:
S
1
=
{
e
4
x
}
,
S
2
=
{
e
2
x
,
e
4
x
,
e
−
4
x
}
,
S
3
=
{
e
4
x
,
e
2
x
}
,
S
4
=
{
e
4
x
,
e
−
4
x
}
,
S
5
=
{
e
4
x
,
7
e
4
x
}
,
S
6
=
{
cosh
4
x
,
sinh
4
x
}
.
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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