I cannot make the mental leap from a vector to a function! In my linear algebra book, it says that a vector is linearly independent if
$\vec V = c1*\vec T_1 + c2*\vec T_2$
And then it goes on to say that 
$y(t) = c1 * e^{-at} + c2*e^{-bt}$
is linearly independent 
My mind cannot comprehend how an analogy can be made in this case. Is there a rigorous theorem some where that says a function is an infinite dimensioned vector? Only then can I completely appreciate linear algebra.
Thanks
 A: It seems that you are somewhat confused about the use of "vectors" terminology.
In the context as mentioned, vectors are quite abstract things. In fact, they
are a generalization of the well known "geometric entities endowed with magnitude
and direction". See the Wikipedia page about 
Vector spaces ,
especially the section about Function spaces . (Note: this answer is partially a dunplicate of Inner product of functions as integration).
A: Vectors $v_1$ and $v_2$ are linearly independent if the only solution to the equation $c_1v_1+c_2v_2=0$ is the solution $c_1=c_2=0$. The functions $e^{-at}$ and $e^{-bt}$ are linearly independent, when considered as vectors, because (assuming $a\ne b$) the only values of $c_1$ and $c_2$ making $c_1e^{-at}+c_2e^{-bt}$ identically zero (that is, zero as a function; zero for all values of $t$) are $c_1=c_2=0$. 
A: Simply put, the official (Bourbaki) definition of a vector is as follows: a vector is an element of a vector space. For instance the elements of $\mathbf{R}^3$ are vectors; but the elements of the $\mathbf{R}$ vector space $\mathscr{F}(\mathbf{F},\mathbf{F})$ of all functions (maps) from $\mathbf{R}$ to $\mathbf{R}$ are also vectors (just because they are elements of a vector space!). One does not need a theorem here, this is merely about having the proper point of view; you only need to absorb the first sentence in this answer.
