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.- 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
$$(a\_1f\_1+a\_2f\_2+a\_3f\_3)(t)=a\_1f\_1(t)+a\_2f\_2(t)+a\_3f\_3(t)=0)$$