3. LetC := {1, cos t, cos 2t,cos 6t} C Vwhere V is the vector space of all real-valued functions defined on the interval (0, 27] as inexample 5 of Section 4.1 on page 204 of the text. Let J = Span(C). Prove that C is a basisfor 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.

The given set is C=1,cost, cos2t, cos3t, …, cos6t⊂V. Where V is the vector space of all real valued…

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.

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.3: Orthonormal Bases:gram-schmidt Process

Problem 17E: Complete Example 2 by verifying that {1,x,x2,x3} is an orthonormal basis for P3 with the inner...

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Question

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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