Linear Algebra - Basis

[discoverer02]

Linear Algebra -- Basis

I had the following problem on an exam this morning and it kind of threw me. I'd appreciate it if someone could review my answers and reasoning and let me know if I answered correctly.

W is a subset of F and spanned by {3, (sinx)^2, (cosx)^2}

a) Prove W is a vector space:

All w's that are members of W can be represented by: a3 + b(sinx)^2 + c(cosx)^2 = f(x)

From this obvious that f(x) is closed under addition and scalar multiplication, so I won't go into details.

b) Find a basis of W:

This is where I was having some problems.

a3 + b(sinx)^2 + c(cosx)^2 = 0;
since (sinx)^2 + (cosx)^2 = 1, there's no linear independance, but is if I calculate the Wronskian of {(sinx)^2, (cosx)^2} is show's they're linearly independent. Because {3, (sinx)^2, (cosx)^2} spanned W, but 3 is a linear combination of {(sinx)^2, (cosx)^2}, {(sinx)^2, (cosx)^2} spans W. Therefore, {(sinx)^2, (cosx)^2} is a basis of W, and the dimension of W is 2.

Is this valid and correct?

Thanks for the help.

 

[turin]

Is this valid and correct?

Looks good to me. Except, I'm a little uneasy how you got the Wronskian to show linear independence.

 

[discoverer02]

Thanks for your reply.

For the Wronskian, W(x) = -Sin(2x). There's an definitely an x where W(x) is not equal to 0, so {(sinx)^2, (cosx)^2} are linearly independent.

 

[turin]

Oh, right. I wasn't paying close enough attention. For some reason I imagined a 1 in there. OK, so now it looks even better to me.