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

[Dustinsfl]

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

Doing the Wronskian here is pain so what other method would be more appropriate?

 

[Mark44]

One of the functions is a linear combination of the other two, so the three functions can't be linearly independent.

 

[Dustinsfl]

How can I go about identifying if one is?

 

[VeeEight]

Use a trig identity for cos2x

 

[Dustinsfl]

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.

 

[VeeEight]

Can cos2x can be written as a linear combination of cos²x and 1?

 

[Dustinsfl]

I see now.

 

[VeeEight]

Great

 

[Mark44]

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.