1. ## Determine a basis

Let S be the vector space consisting of the set of all linear combinations of the functions
f1(x)= e^(x)
f2(x)= e^(-x)
f3(x)= sinh(x)

Determine a basis for S, and hence, find dim[S].
f3(x) = 1/2(f1(x) - f2(x)) so we know that f3(x) is linearly dependent on f1,f2.

Now, we consider f1(x),f2(x)
c1*f1(x) + c2*f2(x) = 0

c1 = c2 = 0. f1 and f2 are Linearly independent and are a basis. Dim[S] = 2.

Thanks

2. Originally Posted by bulldog106
f3(x) = 1/2(f1(x) - f2(x)) so we know that f3(x) is linearly dependent on f1,f2.

Now, we consider f1(x),f2(x)
c1*f1(x) + c2*f2(x) = 0

c1 = c2 = 0. f1 and f2 are Linearly independent and are a basis. Dim[S] = 2.

Thanks
yes, it's correct. although you need to prove that $c_1=c_2=0.$

3. Originally Posted by bulldog106
f3(x) = 1/2(f1(x) - f2(x)) so we know that f3(x) is linearly dependent on f1,f2.

Now, we consider f1(x),f2(x)
c1*f1(x) + c2*f2(x) = 0

c1 = c2 = 0. f1 and f2 are Linearly independent and are a basis. Dim[S] = 2.

If $c1f1(x)+ c2f(x)= c1e^{x}+ c2e^{-x}= 0$ for all x, then it is a constant and so its derivative, $c1e^x- c2e^{-x}= 0$ for all x also. Take x= 0 in both equations.