- #1
davidbenari
- 466
- 18
So when I took linear algebra I was asked to show the well-known properties of orthogonality in the functions cos(mx) and sin(mx)... With this I mean all the usual combinations which I don't see why I should point out explicitly.
The inner product in the vector space was defined ##\int_{-\pi}^{\pi} \phi_n \phi_m dx ## as usual.
What I don't understand is why the space of functions ( cos(mx), sin(nx) ) (with this notation I'm pointing out all cosines and sines of integer angular frequency), is a vector space.
Namely, its not closed under addition! If I add whichever combination I won't get a function of the form cos(mx) or sin(mx) back. There'll always be a phase constant ##\Phi## or something along those lines.
Why are people so wishywashy with this? How can you define an inner product on a space of functions that doesn't form a vector space?
Thanks!
The inner product in the vector space was defined ##\int_{-\pi}^{\pi} \phi_n \phi_m dx ## as usual.
What I don't understand is why the space of functions ( cos(mx), sin(nx) ) (with this notation I'm pointing out all cosines and sines of integer angular frequency), is a vector space.
Namely, its not closed under addition! If I add whichever combination I won't get a function of the form cos(mx) or sin(mx) back. There'll always be a phase constant ##\Phi## or something along those lines.
Why are people so wishywashy with this? How can you define an inner product on a space of functions that doesn't form a vector space?
Thanks!