Let C := {1, cos t, cos 2t, . , cos 6t} C V where V is the vector space of all real-valued functions defined on the interval [0, 27] as in example 5 of Section 4.1 on page 204 of the text. Let J = Span(C). Prove that C is a basis for J.
Let C := {1, cos t, cos 2t, . , cos 6t} C V where V is the vector space of all real-valued functions defined on the interval [0, 27] as in example 5 of Section 4.1 on page 204 of the text. Let J = Span(C). Prove that C is a basis for J.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![3. Let
C := {1, cos t, cos 2t, ...,
cos 6t} C V
where V is the vector space of all real-valued functions defined on the interval [0, 27] as in
example 5 of Section 4.1 on page 204 of the text. Let J = Span(C). Prove that C is a basis
for J.
The following preamble applies to both problems 4 and 5:
Let W C R" be a subspace and let W- C R" be its orthogonal complement. Let S
{u1,..., ' = {v1,. .., vi} be an othogonal basis for W+.
U%} be an orthogonal basis for W and S'
The goal of these exercises is to give two different arguments for the fact that
(*)
dim W + dim W-
= n.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4f523362-59ac-43d1-96c6-e213db7e4e4e%2F0387b63a-79f4-4372-83c5-0810e5a4bea4%2Fyk03ewkp_processed.png&w=3840&q=75)
Transcribed Image Text:3. Let
C := {1, cos t, cos 2t, ...,
cos 6t} C V
where V is the vector space of all real-valued functions defined on the interval [0, 27] as in
example 5 of Section 4.1 on page 204 of the text. Let J = Span(C). Prove that C is a basis
for J.
The following preamble applies to both problems 4 and 5:
Let W C R" be a subspace and let W- C R" be its orthogonal complement. Let S
{u1,..., ' = {v1,. .., vi} be an othogonal basis for W+.
U%} be an orthogonal basis for W and S'
The goal of these exercises is to give two different arguments for the fact that
(*)
dim W + dim W-
= n.
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