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.
Suppose we have some scalars $a\_0,a\_1,a\_2$ in $\Bbb R$ such that
$$a\_0t+a\_1e^t+a\_2\sin t =0 \tag{1}$$
for all real number $t$. Making $t=0$ this gives us $a\_1=0$. Returning to $(1)$, we have
$$a\_0t+a\_2\sin t =0 \tag{2}$$
Now, make $t=\pi$ and then $a\_0\pi=0$, which means $a\_0=0$.