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 $a\_0=0$.
HINT.- A good way to try this problem is to consider that the elements of $V$ are $f\_1,f\_2,f\_3$ where $f\_1(t)=t,f\_2(t)=e^t,f\_3(t)=\sin(t)$ so you have to show that if the linear combination
$$a\_1f\_1+a\_2f\_2+a\_3f\_3=0$$ where les $f\_i$ are three vectors and les $a\_i$ are scalars in your vectorial space, then $a\_1=a\_2=a\_3=0$ (Obviously for each $t$ you have
$$(a\_1f\_1+a\_2f\_2+a\_3f\_3)(t)=a\_1f\_1(t)+a\_2f\_2(t)+a\_3f\_3(t)=0)$$