Question

Find a basis for the subspace of C[0,1] spanned by the points {sin^2 x, cos^2 x, 1}.

Answer 1

Hint: \(\sin^{2}(x) + \cos^{2}(x) = 1\)

Answer 2

I get that they're not Linearly Independent, but how do I "solve" for a basis for this? I get how to do it with Polynomial functions and standard (x,y,z,...n) functions, but the C[0,1] confuses me...

Answer 3

Why would it be any different. Gramm-Schmidt, if nothing else.

Answer 4

That would mean that the basis is just 1 then? But that seems wrong...

Answer 5

Why would it mean that? 1 is a linear combination of the other two. How are the other two dependent?