Linear span of functions Maybe this is something basic but I am not familiar with the term "linear span" and in one question it mentions

the linear span of the functions $f_n(t) = e^{nt}$, $n = 0,1,2,...$ $t \in [a,b]$

What I understood is linear span is the set of linear combination of all elements but I am still unable to understand what that set is.
Thank you very much.
 A: Well, as you said, $f\in \operatorname{span} \{ e^{n t},\ n\in \mathbb{N}\}$ iff there exist $N\in \mathbb{N}$ and $N$ real numbers $\alpha_0, \ldots , \alpha_N$ such that:
$$f(t)=\sum_{n=0}^N \alpha_n\ e^{nt}$$
(this by the very definition of linear combination), therefore:
$$\operatorname{span} \{ e^{n t},\ n\in \mathbb{N}\} =\Bigg\{ \sum_{n=0}^N \alpha_n\ e^{nt},\ N\in \mathbb{N},\ \alpha_0, \ldots , \alpha_N \in \mathbb{R}\Bigg\} \; .$$
A: The linear span of a set $V$ consists of all vectors $w$ of the form
$$w=c_1 v_1+c_2v_2+\cdots+c_n v_n$$ where the $c_i$ are scalars, the $v_i$ are vectors in $V$ and $n$ is a positive integer. 
So, in your case the linear span of the functions $\{f_n\}$ is the set of all finite linear combinations of the $f_n$. An element of the linear span has the form:
$$
\tag{1}f(t)=c_1 e^{n_1t}+c_2 e^{n_2t}+\cdots+ c_m e^{n_m t},
$$
where $m$ is a positive integer, the $n_i$ are non-negative integers, and the $c_i$ are scalars.
In particular, 
$$
f(t)=e^t-2e^{4t}+237 e^{32t}
$$
is in the linear span,
as is
$$
f(t)=3+e^t
$$
$(e^{0t}=1$).
The linear span is the set of all functions of the form in (1).
