Thread: 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, 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.

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 , so the three funtions belong to .

Can you give an explanation ?

Originally Posted by Tala

Can you give an explanation ?

Better ask your concrete doubts.

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

Originally Posted by Tala

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

by definition of the function . On the other hand, (Euler's formula), and from here you can deduce de expression for .

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

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

Now, sum and .

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 ?

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.

Thank you for your help !

Originally Posted by Tala

Thank you for your help !

You are welcome!