This is a homework problem I have and I have the right answer, but I don't know how to do it in proof form . . .
$W$ is the subspace of all continuous real functions spanned by $\{\cos^2(t),\ \sin^2(t),\ \cos(2t)\}$. Find a basis for $W$. I know the given set is linearly dependent, and I know an answer is the right side of the identity of $\cos^2(t) = \sin^2(t) + \cos(2t)$, but I don't know how to show this as a proof.
Believe I found the answer . . .
Create the subset of $W$, $\{\sin^2(t), \cos(2t)\}$, and show that it is linearly independent and that $v_1 +v_2 = v_3$. If both of these conditions hold, which they do for this example, then the subset is a basis for the original set.
Best Answer
Yes, so basically all we have to prove is that the functions $\cos(2t)$ and $\sin^2(t)$ are linearly independent.
Neither of these functions are the zero element (constant $0$ function), so if these were dependent, we must have $\cos(2t)=\lambda\cdot\sin^2(t)$ for a $\lambda$, the equality holds between functions, i.e. holds for all $t$. Now just plug in some values for $t$ to get a contradiction.