Are these two vectors coprime?

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In summary, the conversation discusses the concept of coprime vectors and how to determine if two vectors are coprime. The criteria for proving coprimality is also discussed, along with generalizing the concept to vectors with more than two components. The suggestion is made to move the discussion to the Number Theory forum.
  • #1
phynewb
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Hi guys

I have a question about the coprime of two vectors
For two vectors (x1,x2) and (y1,y2).
Given a,b with gcf (a,b)=1 .i.e. relatively prime.
I do the linear combination of two vectors
a(x1,x2)+b(y1,y2)=n(z1,z2) with some common factor n and gcf(z1,z2)=1.
If n=1 for any a,b, two vectors are said co-prime.
I wonder if any criteria to prove two vectors are coprime.
For example, (2,3),(1,3) are not coprime b/c (2,3)+(1,3)=3(1,2).
But (7,3),(2,1) are coprime b/c a(7,3)+b(2,1)=(7a+2b,3a+b) and gcf(7a+2b,3a+b)=gcf(a,3a+b)=gcf(a,b)=1.
Also how to generalize it to vectors with n components?

Thank you
 
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  • #2
I don't know if there's a name for this operation.
But I can recommend that you move the thread to the Number Theory forum, it seems to belong there more.
 

1. What are co-prime vectors?

Co-prime vectors are two or more vectors that do not share any common factors other than 1. In other words, their components are relatively prime to each other.

2. How are co-prime vectors used in mathematics?

Co-prime vectors are commonly used in number theory and linear algebra. They can be used to represent and solve systems of linear equations, as well as in the study of prime numbers and their properties.

3. What is the significance of co-prime vectors?

Co-prime vectors have a unique property that allows them to span the entire vector space they are in. This makes them useful in many applications, such as cryptography, coding theory, and signal processing.

4. Can co-prime vectors be in any dimension?

Yes, co-prime vectors can exist in any dimension. However, their properties and applications may differ depending on the dimension they are in.

5. How can I determine if two vectors are co-prime?

To determine if two vectors are co-prime, you can calculate their greatest common divisor (GCD) using Euclid's algorithm. If the GCD is 1, then the vectors are co-prime. Alternatively, you can also check if the dot product of the two vectors is equal to 1.

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