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: 
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 $a\_0=0$.
Also you may find if
$$\begin{vmatrix} f\_1(t) & f\_2(t) & f\_3(t) \\ f\_1'(t) & f\_2'(t) & f\_3'(t)\\ f\_1''(t) & f\_2''(t) & f\_3''(t)\end{vmatrix}= 0,$$ then the three functions are linearly dependent (LD) otherwise they are linearly independent (LI).