Linear algebra- set of complex valued functions

In summary, the function f in the domain V is real valued if and only if f(-t)=f(t) for all t in R, and is not real-valued if and only if f(t) is not conjugate(f(-t)).
  • #1
gotmilk04
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0

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.


Homework Equations





The Attempt at a Solution


Not quite sure what this means, just need a place to start really.
 
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  • #2
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)?
 
  • #3
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?
 
  • #4
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)?
 
  • #5
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)?
 
  • #6
Yes, so u(-t)+iv(-t)=u(t)-iv(t), right?
 
  • #7
Yep. Now I'm not sure where to go from here.
 
  • #8
Equate real and imaginary parts of both sides. Come on, help me out here.
 
  • #9
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)?
 
  • #10
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?
 
  • #11
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?
 
  • #12
Yes, u(t)=1 and v(t)=t works too. If you want to make it even simpler.
 
  • #13
Ah yes, that is much simpler.
Thanks so much for the help, sorry I was very lost before. I appreciate your patience and guidance!
 

1. What is linear algebra?

Linear algebra is a branch of mathematics that deals with the study of vector spaces and linear transformations between those spaces. It involves the use of matrices, vectors, and systems of linear equations to solve problems related to geometry, physics, and engineering.

2. What are complex valued functions?

Complex valued functions are functions that take complex numbers as inputs and produce complex numbers as outputs. These functions are typically denoted by f(z) and can be represented by an expression such as f(z) = z^2 + 2z + 1, where z is a complex number.

3. How is linear algebra used in studying complex valued functions?

Linear algebra is used to analyze and manipulate complex valued functions, as well as to solve systems of equations involving these functions. It also plays a crucial role in understanding the properties and behavior of complex valued functions, such as continuity, differentiability, and convergence.

4. Can complex valued functions be visualized?

Yes, complex valued functions can be visualized using a complex plane. In this plane, the horizontal axis represents the real part of a complex number, and the vertical axis represents the imaginary part. This allows us to graphically represent and study the behavior of complex valued functions.

5. What are some real-life applications of linear algebra and complex valued functions?

Linear algebra and complex valued functions have numerous real-life applications, including image processing, signal processing, quantum mechanics, and electrical engineering. They are also used in fields such as computer graphics, cryptography, and finance.

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