1. Subspaces and functions

Hi

The complex functions z(t), t
∈ R, can be differentiated a number of times and are a vector space V :

W = span(e-t, e-it, et, eit) and W is a subspace of V.

Which of the following 3 functions belong to W ?

z1(t) = 3et, z2(t) = cos(t), z3(t) = sinh(t)

If the functions were vectors for example like this W= span((1,2),(2,1)) and I had to determine if the vector (-2,3) belonged to W, I would solve the inhomogeneous linear system but since it's not I have no idea how to determine this.

I hope that somebody can give me a hint.

2. Re: Subspaces and functions

Originally Posted by Tala
Which of the following 3 functions belong to W ? z1(t) = 3et, z2(t) = cos(t), z3(t) = sinh(t)
We have $z_1(t)=3e^{t},\;z_2(t)=\dfrac{1}{2}e^{it}+\dfrac{1 }{2}e^{-it},\;z_3(t)=\dfrac{1}{2}e^{t}-\dfrac{1}{2}e^{-t}$, so the three funtions belong to $W$.

3. Re: Subspaces and functions

Can you give an explanation ?

4. Re: Subspaces and functions

Originally Posted by Tala
Can you give an explanation ?

5. Re: Subspaces and functions

I don't understand how you get z2 to that form ?

6. Re: Subspaces and functions

Originally Posted by Tala
I don't understand how you get z2 and z3 to that form ?
$\sinh t=\frac{e^t-e^{-t}}{2}$ by definition of the function $\sinh$. On the other hand, $e^{it}=\cos t+i\sin t$ (Euler's formula), and from here you can deduce de expression for $z_2(t)$.

7. Re: Subspaces and functions

This might be stupid but I don't get your z2 expression when I isolate cost in e^it = cost + i*sint ?

8. Re: Subspaces and functions

Originally Posted by Tala
This might be stupid but I don't get your z2 expression when I isolate cost in e^it = cost + i*sint ?
Don't worry. We have $\begin{Bmatrix} e^{it}=\cos t+i\sin t\quad (1)\\e^{-it}=\cos t-i\sin t\quad (2)\end{matrix}$

Now, sum $(1)$ and $(2)$.

9. Re: Subspaces and functions

I see ! That makes sense.
And one last question the functions belong to W because they each can be written as a linear combination of span(e-t, e-it, et, eit) right ?

10. Re: Subspaces and functions

Originally Posted by Tala
I see ! That makes sense.
And one last question the functions belong to W because they each can be written as a linear combination of span(e-t, e-it, et, eit) right ?
Well, as linear combination of those four functions.

11. Re: Subspaces and functions

Thank you for your help !

12. Re: Subspaces and functions

Originally Posted by Tala
Thank you for your help !
You are welcome!