How to Determine the Dimension of Span(1, cos(2x), cos²(x)) in C[-π, π]?

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Homework Help Overview

The discussion revolves around determining the dimension of the span of the functions 1, cos(2x), and cos²(x) within the space C[-π, π]. Participants are exploring the linear independence of these functions.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants discuss the use of the Wronskian and consider alternative methods for assessing linear independence. There is a focus on identifying relationships between the functions, particularly through trigonometric identities.

Discussion Status

Some participants have pointed out that one function may be a linear combination of the others, prompting further exploration of this idea. There is an acknowledgment of the complexity involved in establishing linear independence.

Contextual Notes

Participants are grappling with the implications of trigonometric identities and their role in determining linear combinations, as well as the challenge of using the Wronskian method.

Dustinsfl
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C[-pi, pi]
Span(1, cos (2x), cos2 (x))

Doing the Wronskian here is pain so what other method would be more appropriate?
 
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One of the functions is a linear combination of the other two, so the three functions can't be linearly independent.
 
How can I go about identifying if one is?
 
Use a trig identity for cos2x
 
cos2x=2cos^2 x-1 but I don't see how that is a multiple of cos2x sine it doesn't = cos^2 x outright.
 
Can cos2x can be written as a linear combination of cos2x and 1?
 
I see now.
 
Great :smile:
 
Dustinsfl said:
cos2x=2cos^2 x-1 but I don't see how that is a multiple of cos2x sine it doesn't = cos^2 x outright.
I realize that you already figured this out, but if you look at what I said, I didn't say that one of the functions was a multiple of another. I said that one function was a linear combination of the other two.
 

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