# Basis for a vector space consisting of the set of linear combinations of these functions

S is the vector space consisting of the set of all linear combinations of the functions $f_1(x)=e^x$, $f_2(x)=e^{-x}$, $f_3=sinh(x)$. Find a basis for S and find the dim[S].

First, I let

$c_1f_1$+$c_2f_2$+$c_3f_3$=$f$

Then, since $sinh(x) =\frac{e^x-e^{-x}}{2}$,

$f_3=\frac{f_1-f_2}{2}$

Substituting back in, I can get

$c_1f_1$+$c_2f_2$+$c_3\frac{f_1-f_2}{2}$ =$f$

$(c_1+\frac{1}{2}c_3)f_1$+$(c_2-\frac{1}{2}c_3)f_2$ =$f$

Am I on to something?

• Nice name, by the way. :) Nov 29 '14 at 17:58

Yup! :) You are correct. You have shown that $f_3$ is a linear combination of the other two basis vectors, and thus not needed to form the basis of the vector space (although you could have just as easily proved any one of the three given functions wasn't needed). As a result, since you have two basis functions, your subspace dimension is 2.
$f_1$ and $f_2$ are linearly independent and, as you point out, $f_3$ is dependent on these two, so a basis is $B=\{f_1,f_2\}$ and the dimension is 2.
$f_1$ and $f_2$ are linearly independent (check using the definition), while $f_3$ is a linear combination of the first two. So the basis (i.e. linearly independent set which spans the space) is just $\{f_1,f_2\}$. Since this set contains two elements, the space is two dimensional.