I'm trying to show that the functions $c_1 + c_2 \sin^2 x + c_3 \cos^2 x$ forms a vector space.
And I will need to find a basis of it, and its dimension.
Is there a way how to do this without verifying the 8 axioms for a vector space, and if we let the set $X = \{c_1 + c_2 \sin^2 x + c_3 \cos^2 x\}$ then we note that $1 = \sin^2 x + \cos^2 x$, and this is enough. So the dimension is $2$. Thanks.
Can you please provide clarification on how the argument of the subspace of the vector space follows? I think you did it already by inspection, but its not very complete to me, can you please write it down? Thanks