Subspaces and functions

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November 10th 2012, 10:54 AM
Tala

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Subspaces and functions

Hi

The complex functions z(t), t ∈ R, can be differentiated a number of times and are a vector space V :
Attachment 25625
W = span(e^-t, e^-it, e^t, e^it) and W is a subspace of V.

Which of the following 3 functions belong to W ?

z₁(t) = 3e^t, 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.
Sorry for the bad description.
November 10th 2012, 11:11 AM
FernandoRevilla

Re: Subspaces and functions

Quote:

Originally Posted by Tala

Which of the following 3 functions belong to W ? z₁(t) = 3e^t, 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 $W$ .
November 10th 2012, 11:13 AM
Tala

Re: Subspaces and functions

Can you give an explanation ?
November 10th 2012, 11:15 AM
FernandoRevilla

Re: Subspaces and functions

Quote:

Originally Posted by Tala

Can you give an explanation ?

Better ask your concrete doubts.
November 10th 2012, 11:25 AM
Tala

Re: Subspaces and functions

I don't understand how you get z2 to that form ?
November 10th 2012, 11:32 AM
FernandoRevilla

Re: Subspaces and functions

Quote:

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 $z_2(t)$ .
November 10th 2012, 11:41 AM
Tala

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 ?
November 10th 2012, 11:46 AM
FernandoRevilla

Re: Subspaces and functions

Quote:

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 $(2)$ .
November 10th 2012, 11:53 AM
Tala

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, e^t, e^it) right ?
November 10th 2012, 11:56 AM
FernandoRevilla

Re: Subspaces and functions

Quote:

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, e^t, e^it) right ?

Well, as linear combination of those four functions.
November 10th 2012, 12:00 PM
Tala

Re: Subspaces and functions

Thank you for your help !
November 10th 2012, 12:05 PM
FernandoRevilla

Re: Subspaces and functions

Quote:

Originally Posted by Tala

Thank you for your help !

You are welcome! :)