# Linear algebra- set of complex valued functions

## Homework Statement

Let V be the set of all complex-valued functions f on the real line such that (for all t in R),
f(-t)=f(t) with a bar on top (can't figure out Latex, sorry)
The bar denotes complex conjugation.

Give an example of a function in V which is not real-valued.

## The Attempt at a Solution

Not quite sure what this means, just need a place to start really.

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Dick
Homework Helper
Write f(t)=u(t)+iv(t) where u and v are real valued functions. Equate the real and imaginary parts of both sides of your equation. What are the conditions on u(t) and v(t)?

I didn't really know there were conditions on u(t) and v(t).
To get an example, can I just make those fuctions whatever real valued function that I want, like t-2, and then plug it in?

Dick
Homework Helper
I mean the conditions you derive from f(-t)=conjugate(f(t)). If f(t)=u(t)+iv(t) isn't conjugate(f(t))=u(t)-iv(t)?

Ohh, yeah, it does. So I have to make f(t) so that when the t's are (-t)'s, u(t)+iv(t) turns into u(t)-iv(t)?

Dick
Homework Helper
Yes, so u(-t)+iv(-t)=u(t)-iv(t), right?

Yep. Now I'm not sure where to go from here.

Dick
Homework Helper
Equate real and imaginary parts of both sides. Come on, help me out here.

I'm sorry, I'm very confused here and I've never learned this before, so it's very frustrating.

When you equate the real and imaginary parts, do you get
u(-t)=u(t) and v(-t)=-v(t)?

Dick
Homework Helper
Yes, exactly. Can you find two functions u and v so that u(-t)=u(t) and v(-t)=(-v(t)) and v is not equal to zero? So f is not real valued?

So if u(t)=t^2+1 and v(t)=t^3-t,
then f(t)=t^2 + 1 + i(t ^3-t)
= it^3 + t^2 - it + 1
Which is not real valued, so that's an example, correct?

Dick