There is a vector space with set F, of all real functions. It has the usual operations of addition of functions and multiplication by scalars. You have to determine whether this equation is a subspace of F: [tex]f''(x) + 3f'(x) + x^2 f(x) = sin(x)[/tex]
[tex]f''(x) + 3f'(x) + x^2 f(x) = sin(x)[/tex] the 0 vector/function
The Attempt at a Solution
So, to test that it is non-empty set I used the 0 vector/function. However, is this the same as letting "x=0"? If so, it would then be:
[tex]f''(0) + 3f'(0) + x^2 f(0) = sin(0)[/tex] and therefore [tex]0 = 0[/tex] proving that the set is non-empty.
As, wouldn't it be what value also makes sin(x) = 0 (which is x=0) and so, would this be correct?
I just want to clarify whether it is before I continue further with solving the problem.