I want to determine whether 3 functions are linearly independent:
\begin{align*}
x_1(t) & = 3 \\
x_2(t) & = 3\sin^2(t) \\
x_3(t) & = 4\cos^2(t)
\end{align*}
Definition of Linear Independence: $c_1x_1 + c_2x_2 + c_3x_3 = 0 \implies c_1=c_2=c_3=0$ (only the trivial solution)
So we have:
\begin{align}
3c_1 + 3c_2\sin^2(t) + 4c_3\cos^2(t) = 0
\end{align}
My first idea is to differentiate both sides and get:
$6c_2\sin(t)\cos(t) – 8c_3\cos(t)\sin(t) = 0$
Then we can factor to get:
$\sin(t)\cos(t)(6c_2 – 8c_3) = 0$
So $c_3= \dfrac{6}{8}c_2$ gives the equation equals zero. Thus all $c$ are not $0$ and thus $x_1, x_2, x_3$ are linearly dependent.
Is this correct? Or is there a cleaner way to do this?
Best Answer
Yes, indeed, your answer is fine. And it would have been a particularly fine determining the linear (in)dependence of a system of equations that doesn't readily admit of another observation about the relationship between $\cos^2 t$ and $\sin^2 t$ $(\dagger)$. Indeed, you're one step away from working with the Wronskian, which is a useful tool to prove linear independence.
$(\dagger)$ Now, to the observation previously noted: You could have also used the fact that $$x_1(t) - \left[(x_2(t) +\frac 34 x_3(t)\right] = 3 - (3 \sin^2 t + 3\cos^2 t)= 3 - 3\left(\underbrace{\sin^2(t) + \cos^2(t)}_{\large = 1}\right)=0$$
and saved yourself a little bit of work: you can read off the nonzero coefficients $c_i$ to demonstrate their existence: $c_1 = 1, c_2 = -1, c_3 = -\frac 34$, or you could simply express $x_1$ as a linear combination of $x_2, x_3$, to conclude the linear dependence of the vectors. (But don't count on just any random set of vectors turning out so nicely!)