Proving V is a Real Vector Space Given V is a Complex Vector Space

In summary, to prove that if V is a complex vector space, then V is also a real vector space, you must first define a new scalar multiplication operation using the given definitions. Then, you can use this new operation to show that all the vector space axioms hold for the set of vectors in V with real scalars, thus proving that V is also a real vector space.
  • #1
bendaddy
4
0

Homework Statement


Hello, I'm having a little difficulty with this proof.
Prove: If V is a complex vector space, then V is also a real vector space.


Homework Equations


Definition 1: A vector space V is called a real vector space if the scalars are real numbers.
Definition 2: A vector space V is called a complex vector space if the scalars are complex numbers.


The Attempt at a Solution


I've tried a proof by contradiction saying that "V is a complex vector space and V is not a real vector space." Can't seem to find any sort of contradiction throughout my argument though.
 
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  • #2
It's good that you didn't find any contradictions, because there aren't any. The set of vectors in a complex vector space with complex scalars are also a vector space over the reals. Just try to prove it directly. What things do you need to prove?
 
  • #3
I just need to develop a proof using the vector space axioms and the two definitions listed above that if V is a complex vector space, then V is also a real vector space. Not sure how to go about proving it directly though...
 
  • #4
bendaddy said:
I just need to develop a proof using the vector space axioms and the two definitions listed above that if V is a complex vector space, then V is also a real vector space. Not sure how to go about proving it directly though...

State the axioms you have to prove. Take them one by one.
 
  • #5
Before you can check any of the vector space axioms, you must define a new scalar multiplication operation. The one you have is a map from ℂ×V into V. You need to use it to define a map from ℝ×V into V.
 

FAQ: Proving V is a Real Vector Space Given V is a Complex Vector Space

1. What is a complex vector space?

A complex vector space is a mathematical structure that consists of a set of elements called vectors, along with rules for addition and scalar multiplication. The vectors in a complex vector space have complex entries, meaning they contain both real and imaginary parts. This allows for more complex and powerful calculations than in a real vector space.

2. How is a complex vector space different from a real vector space?

The main difference between a complex vector space and a real vector space is the type of numbers used for the entries of the vectors. In a real vector space, the entries are real numbers, while in a complex vector space, the entries are complex numbers. This allows for more flexibility and a wider range of operations in a complex vector space.

3. What are some examples of complex vector spaces?

Some examples of complex vector spaces include the space of complex-valued polynomials, the space of complex-valued matrices, and the space of complex-valued functions. These spaces are commonly used in mathematics, physics, and engineering for their ability to model complex systems and phenomena.

4. What is the dimension of a complex vector space?

The dimension of a complex vector space is the number of vectors in a basis for that space. Similar to a real vector space, the dimension of a complex vector space can be finite or infinite. If it is finite, the dimension is simply the number of vectors in the basis. If it is infinite, the dimension is called the cardinality of the basis.

5. How are complex vector spaces used in real-world applications?

Complex vector spaces have a wide range of applications in fields such as physics, engineering, and data analysis. They are used to model and understand complex systems and phenomena that involve both real and imaginary quantities. Some examples of real-world applications of complex vector spaces include quantum mechanics, signal processing, and image processing.

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