Re: Subspaces and functions

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**Tala** Which of the following 3 functions belong to W ? z_{1}(t) = 3e^{t}, z_{2}(t) = cos(t), z_{3}(t) = sinh(t)

We have $\displaystyle z_1(t)=3e^{t},\;z_2(t)=\dfrac{1}{2}e^{it}+\dfrac{1 }{2}e^{-it},\;z_3(t)=\dfrac{1}{2}e^{t}-\dfrac{1}{2}e^{-t}$, so the three funtions belong to $\displaystyle W$.

Re: Subspaces and functions

Can you give an explanation ?

Re: Subspaces and functions

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**Tala** Can you give an explanation ?

Better ask your concrete doubts.

Re: Subspaces and functions

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

Re: Subspaces and functions

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**Tala** I don't understand how you get z2 and z3 to that form ?

$\displaystyle \sinh t=\frac{e^t-e^{-t}}{2}$ by definition of the function $\displaystyle \sinh $. On the other hand, $\displaystyle e^{it}=\cos t+i\sin t$ (Euler's formula), and from here you can deduce de expression for $\displaystyle z_2(t)$.

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 ?

Re: Subspaces and functions

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**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 $\displaystyle \begin{Bmatrix} e^{it}=\cos t+i\sin t\quad (1)\\e^{-it}=\cos t-i\sin t\quad (2)\end{matrix}$

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

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 ?

Re: Subspaces and functions

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**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.

Re: Subspaces and functions

Thank you for your help !

Re: Subspaces and functions

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**Tala** Thank you for your help !

You are welcome! :)