I am having trouble understanding the span of functions, my problem is:
What is $\operatorname{span}\{a\sin(x),b\}$, and what is its dimension?
I understand this in terms of vectors, but not in terms of functions, so any enlightenment would be very much appreciated.
Enlightenment: Functions are vectors.
Assuming you mean $a$ and $b$ are fixed constants in your base field $F$ (probably $\mathbb{R}$ or $\mathbb{C}$), then $$ \text{Span}_F\{a\sin x,b\} = \{f_1 a\sin x + f_2 b : f_1,f_2 \in F\}. $$
When viewing $v = a \sin x$ and $w = b$ as vectors, the equation $$ f_1 a\sin x + f_2 b = f_1 v + f_2 w = 0 $$ means that for all values of $x \in F$ (or whatever the domain of $\sin x$ is assumed to be) $$ f_1 a\sin x + f_2 b = 0. $$ Are there any constants $f_1$ and $f_2$ that make this true? If not, then the span has dimension $2$. If so, its dimension is less than two, since this says $v$ and $w$ are linearly dependent.