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. 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. 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 illustrates how the set B=leftex is a basis for V.
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.
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.
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