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.
Best Answer
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).