Solution Found!
Assume the following trigonometric identities (see
Chapter 4, Problem 34E(choose chapter or problem)
Problem 34E
Assume the following trigonometric identities (see Exercise 37 in Section 4.1).
Let H be the subspace of functions spanned by the functions in . Then is a basis for H, by Exercise 38 in Section 4.3.
a. Write the -coordinate vectors of the vectors in C, and use them to show that C is a linearly independent set in H.
b. Explain why C is a basis for H.
Reference 37 in Section 4.1:
[M] The vector space
contains at least two interesting functions that will be used in a later exercise:
Study the graph of f for , and guess a simple formula for f (t). Verify your conjecture by graphing the difference between 1 + f (t) and your formula for f (t) (Hopefully, you will see the constant function 1.) Repeat for g.
Reference 38 in Section 4.3:
[M] Show that is a linearly independent
set of functions defined on . Use the method of Exercise 37. (This result will be needed in Exercise 34 in Section 4.5.)
Questions & Answers
QUESTION:
Problem 34E
Assume the following trigonometric identities (see Exercise 37 in Section 4.1).
Let H be the subspace of functions spanned by the functions in . Then is a basis for H, by Exercise 38 in Section 4.3.
a. Write the -coordinate vectors of the vectors in C, and use them to show that C is a linearly independent set in H.
b. Explain why C is a basis for H.
Reference 37 in Section 4.1:
[M] The vector space
contains at least two interesting functions that will be used in a later exercise:
Study the graph of f for , and guess a simple formula for f (t). Verify your conjecture by graphing the difference between 1 + f (t) and your formula for f (t) (Hopefully, you will see the constant function 1.) Repeat for g.
Reference 38 in Section 4.3:
[M] Show that is a linearly independent
set of functions defined on . Use the method of Exercise 37. (This result will be needed in Exercise 34 in Section 4.5.)
ANSWER:
Solution 34E
Step 1 of 6
Let the basis and
And also consider the following trigonometric identities:
Every element is function from
And is subspace of vector space of all functions from
The set is linearly independent set in vector space of all functions from
So, the set is basis of subspace.
(a)
Objective is to write coordinate vectors of the vectors in
The of each function of is
