8. [M] Show that {1, cos t, cos? t,..., cos t} is a linearly inde- pendent set of functions defined on R. Use the method of Exercise 37. (This result will be needed in Exercise 34 in Section 4.5.)
8. [M] Show that {1, cos t, cos? t,..., cos t} is a linearly inde- pendent set of functions defined on R. Use the method of Exercise 37. (This result will be needed in Exercise 34 in Section 4.5.)
Linear Algebra: A Modern Introduction
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Chapter5: Orthogonality
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Problem 6AEXP
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Question
Number 38 please
![and let
2
GEL
2
4
U2 =
Uz =
-1
1
V, then
depen-
sforma-
1
-4
2.
-2
3
4
dent set
V1 =
-1
V2 =
V3 =
ndepen-
3
-6
Find bases for H, K, and H + K. (See Exercises 33 and 34
in Section 4.1.)
that an
y that if
mendent,
ws that
adеpen-
nis case
37. [M] Show that {t, sin t, cos 2t, sin t cos t} is a linearly inde-
pendent set of functions defined on R. Start by assuming that
C i + C2 sin t + C3 cos 2t + C4 sint cos t 0
(5)
Equation (5) must hold for all real t, so choose several
specific values of t (say, t 0, .1, .2) until you get a system
of enough equations to determine that all the c¡ must be zero.
=1-
ec;
or why
38. [M] Show that {1, cos t, cos² t,..., cos t} is a linearly inde-
pendent set of functions defined on R. Use the method of
Exercise 37. (This result will be needed in Exercise 34 in
Section 4.5.)
-t, and
depen-
WEB
PRACTICE PROBLEMS
V2]. Row operations show that
-2
-2
A =
-2
0.
3
3
-9
0.
of A contains a pivot position. So the columns of A do not span R',
in Section 1.4. Hence {v1, v2} is not a basis for R. Since v and
2, they cannot possibly be a basis for R2. However, since vi and v2
inearly independent, they are a basis for a subspace of R, namely,
A whose column space is the space spanned by {V1, V2, V3, V4}, and
e A to find its pivot columns.
6.
2 -4
1
-4
2 -4
2 -2 -8
0.
20
4 -20
1-5
2.
45](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fee22cf2f-b974-4b00-a3cf-09b388e7d65d%2Fa40386a6-705c-41d2-89f7-208c248a3979%2F4ax1dg3_processed.jpeg&w=3840&q=75)
Transcribed Image Text:and let
2
GEL
2
4
U2 =
Uz =
-1
1
V, then
depen-
sforma-
1
-4
2.
-2
3
4
dent set
V1 =
-1
V2 =
V3 =
ndepen-
3
-6
Find bases for H, K, and H + K. (See Exercises 33 and 34
in Section 4.1.)
that an
y that if
mendent,
ws that
adеpen-
nis case
37. [M] Show that {t, sin t, cos 2t, sin t cos t} is a linearly inde-
pendent set of functions defined on R. Start by assuming that
C i + C2 sin t + C3 cos 2t + C4 sint cos t 0
(5)
Equation (5) must hold for all real t, so choose several
specific values of t (say, t 0, .1, .2) until you get a system
of enough equations to determine that all the c¡ must be zero.
=1-
ec;
or why
38. [M] Show that {1, cos t, cos² t,..., cos t} is a linearly inde-
pendent set of functions defined on R. Use the method of
Exercise 37. (This result will be needed in Exercise 34 in
Section 4.5.)
-t, and
depen-
WEB
PRACTICE PROBLEMS
V2]. Row operations show that
-2
-2
A =
-2
0.
3
3
-9
0.
of A contains a pivot position. So the columns of A do not span R',
in Section 1.4. Hence {v1, v2} is not a basis for R. Since v and
2, they cannot possibly be a basis for R2. However, since vi and v2
inearly independent, they are a basis for a subspace of R, namely,
A whose column space is the space spanned by {V1, V2, V3, V4}, and
e A to find its pivot columns.
6.
2 -4
1
-4
2 -4
2 -2 -8
0.
20
4 -20
1-5
2.
45
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