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,..., uk} be an orthogonal basis for W and S' = {v1,.., Vi} be an othogonal basis for W-.
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%2F5cfad234-a2c2-4215-85c9-d792d8016066%2F54072190-0176-4fa0-99b3-fd25ab921b55%2F9vfunkk_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,..., uk} be an orthogonal basis for W and S' = {v1,.., Vi} be an othogonal basis for W-.
The goal of these exercises is to give two different arguments for the fact that
(*)
dim W + dim W-
=n.
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