The set { sin x , cos x } is a subset of the vector space C [ − π , π ] . Prove that the set is linearly independent. [ Hint: Set f ( x ) = c 1 sin x + c 2 cos x and assume that f ( x ) = θ ( x ) . Then f ( 0 ) = 0 and f ( π / 2 ) = 0 .]
The set { sin x , cos x } is a subset of the vector space C [ − π , π ] . Prove that the set is linearly independent. [ Hint: Set f ( x ) = c 1 sin x + c 2 cos x and assume that f ( x ) = θ ( x ) . Then f ( 0 ) = 0 and f ( π / 2 ) = 0 .]
Solution Summary: The author explains how the set leftmathrmsinx,macrosnright is linearly independent.
The set
{
sin
x
,
cos
x
}
is a subset of the vector space
C
[
−
π
,
π
]
. Prove that the set is linearly independent. [Hint: Set
f
(
x
)
=
c
1
sin
x
+
c
2
cos
x
and assume that
f
(
x
)
=
θ
(
x
)
. Then
f
(
0
)
=
0
and
f
(
π
/
2
)
=
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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