In Exercises 34 and 35 , V is the set of functions V = { f ( x ) : f ( x ) = a e x + b e 2 x + c e 3 x + d e 4 x for real numbers a , b , c , d } . It can be shown that V is a vector space. Let S = { g 1 ( x ) , g 2 ( x ) , g 3 ( x ) } be the subset of V , where g 1 ( x ) = e x − e 4 x , g 2 ( x ) = e 2 x + e 3 x , and g 1 ( x ) = − e x + e 3 x + e 4 x .use Theorem 5 and basis B of exercise 34 to show that S is a linearly independent set. 3 4 . Show that B = { e x , e 2 x , e 3 x , e 4 x } is a basis for V . [ Hint: To see that B is a linearly independent set, let h ( x ) = c 1 e x + c 2 e 2 x + c 3 e 3 x + c 4 e 4 x and assume that h ( x ) = θ ( x ) . Then h ′ ( x ) = θ ( x ) , h ′ ′ ( x ) = θ ( x ) , and h ′ ′ ′ ( x ) = θ ( x ) . Therefore, h ( 0 ) = 0 , h ′ ( 0 ) = 0 , h ′ ′ ( 0 ) = 0 , and h ′ ′ ′ ( 0 ) = 0 .]
In Exercises 34 and 35 , V is the set of functions V = { f ( x ) : f ( x ) = a e x + b e 2 x + c e 3 x + d e 4 x for real numbers a , b , c , d } . It can be shown that V is a vector space. Let S = { g 1 ( x ) , g 2 ( x ) , g 3 ( x ) } be the subset of V , where g 1 ( x ) = e x − e 4 x , g 2 ( x ) = e 2 x + e 3 x , and g 1 ( x ) = − e x + e 3 x + e 4 x .use Theorem 5 and basis B of exercise 34 to show that S is a linearly independent set. 3 4 . Show that B = { e x , e 2 x , e 3 x , e 4 x } is a basis for V . [ Hint: To see that B is a linearly independent set, let h ( x ) = c 1 e x + c 2 e 2 x + c 3 e 3 x + c 4 e 4 x and assume that h ( x ) = θ ( x ) . Then h ′ ( x ) = θ ( x ) , h ′ ′ ( x ) = θ ( x ) , and h ′ ′ ′ ( x ) = θ ( x ) . Therefore, h ( 0 ) = 0 , h ′ ( 0 ) = 0 , h ′ ′ ( 0 ) = 0 , and h ′ ′ ′ ( 0 ) = 0 .]
Solution Summary: The author explains that the set S is a linearly independent set.
In Exercises
34
and
35
,
V
is the set of functions
V
=
{
f
(
x
)
:
f
(
x
)
=
a
e
x
+
b
e
2
x
+
c
e
3
x
+
d
e
4
x
for real numbers
a
,
b
,
c
,
d
}
.
It can be shown that
V
is a vector space.
Let
S
=
{
g
1
(
x
)
,
g
2
(
x
)
,
g
3
(
x
)
}
be the subset of
V
, where
g
1
(
x
)
=
e
x
−
e
4
x
,
g
2
(
x
)
=
e
2
x
+
e
3
x
, and
g
1
(
x
)
=
−
e
x
+
e
3
x
+
e
4
x
.use Theorem
5
and basis
B
of exercise
34
to show that
S
is a linearly independent set.
3
4
.
Show that
B
=
{
e
x
,
e
2
x
,
e
3
x
,
e
4
x
}
is a basis for
V
. [Hint: To see that
B
is a linearly independent set, let
h
(
x
)
=
c
1
e
x
+
c
2
e
2
x
+
c
3
e
3
x
+
c
4
e
4
x
and assume that
h
(
x
)
=
θ
(
x
)
. Then
h
′
(
x
)
=
θ
(
x
)
,
h
′
′
(
x
)
=
θ
(
x
)
, and
h
′
′
′
(
x
)
=
θ
(
x
)
. Therefore,
h
(
0
)
=
0
,
h
′
(
0
)
=
0
,
h
′
′
(
0
)
=
0
, and
h
′
′
′
(
0
)
=
0
.]
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.
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.