Question: Is $S$ linearly dependent on $\textsf V = \mathcal{F}(\Bbb R,\Bbb R)$ and $S = \{t, e^t ,\sin(t)\}$.
How to prove that a set of functions are linearly dependent?

Answer: 
Here is a another way to skin the cat:

Suppose $\alpha\_1 t + \alpha\_2 e^t + \alpha\_3 \sin t = 0$ for all $t$.

If we differentiate twice and set $t = 0$, we get $\alpha\_2 e^{\pi \over 2} = 0$ and so $\alpha\_2 = 0$.

If we differentiate twice and set $t = -{\pi \over 2}$, we get $\alpha\_3 = 0$.

Finally, set $t=1$ to get $\alpha\_1 = 0$.

Hence $S$ is linearly independent.
*Hint.* Write one as a linear combination of the others. That is, look for real numbers $a,b,c$ not simultaneously vanishing, so that we may write $$at+be^t+c\sin t=0.$$