How to prove a function is vector valued

In summary, the conversation discusses the proof of a function being vector or complex valued, depending on its input variables. The co-domain of the function is determined by its definition and the assumption is made that the input variables are real-valued and belong to the vector space R2. It is also mentioned that for a function to be complex valued, its input variables must be separately real-valued.
  • #1
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Suppose you have a function f. Perhaps its input is one variable, f(x), or maybe its input is two variables, f(x,y). How can you prove that the function is itself vector valued,
f=(f1(x,y),f2(x,y))?

Or perhaps it would be easier to prove that f is a complex valued function,
f=f1(x,y)+if2(x,y).
 
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  • #2
All I can say is that the way it is defined tells what the "co-domain" is. I presume you are assuming that f1(x,y) and f2(x,y) are themselves real-valued and that (x, y) is a member of the vector space R2. However, R2, alone, is not "given" as a vector space. You would have to assume, or be given, the standard definitions for sum of two pairs and scalar multiplication.

Similarly, if f1(x,y) and f2(x,y) are real valued, then f1(x,y)+if2(x,y) is, by definition, a complex number for every (x, y). (The requirment that f1 and f2, separately, be real valued is necessary. For example, if f1(x, y)= x+ y and f2(x, y)= i, f(x,y)= f1(x,y)+ if(x,y)= x+ y- 1 which is real valued.)
 

1. How do I know if a function is vector valued?

A function is considered vector valued if its output is a vector, meaning it has both magnitude and direction. This can be identified by looking at the function's domain and range. If both the input and output of the function are vectors, then it is vector valued.

2. What are the key properties of a vector valued function?

A vector valued function must satisfy two key properties: the output must be a vector, and the input must be a vector. Additionally, the function must follow the rules of vector operations, such as vector addition and scalar multiplication.

3. How do I prove that a function is vector valued?

To prove that a function is vector valued, you must show that its output is a vector and its input is a vector. You can do this by using the definition of a vector valued function and showing that the function satisfies the key properties mentioned above.

4. Can a function be both scalar and vector valued?

Yes, a function can be both scalar and vector valued. This means that the function's output can be a combination of both scalars and vectors, depending on the input. For example, a function that takes in a vector and outputs a scalar is both scalar and vector valued.

5. How can I use vector valued functions in real-life applications?

Vector valued functions have many real-life applications, such as in physics and engineering. They can be used to model the motion of objects, describe forces and velocities, and solve systems of equations. They are also used in computer graphics and animation to create 3D images and movements.

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