Understanding Odd and Even Permutations in Linear Algebra - Quick Help

In summary, the conversation is about determining whether a permutation is odd or even in Linear Algebra. The method for determining this is to count the "steps" needed for the permutation and see if the number is odd or even. An even permutation is the product of an even number of 2-cycles, while an odd permutation is not. The example (1 2 3 4) is odd, while (1 3)(2 4) is even.
  • #1
elle
91
0
Linear Algebra help -urgent!

Hi,
I'm doing a bit of self learning on Linear Algebra but I'm kinda confused with the information given on odd and even permutations?

How do you determine whether a permutation is odd or even?

For example, the book has given some examples such as:

(1 2 3 4) is odd, (1 3)(2 4) is even? :confused:

Can someone help please? Thank you!
 
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  • #2
Try to count the 'steps' you need to do a permutation and see if the number is odd or even.
 
  • #3
A permutation is even if it's the product of an even number of 2-cycles. That's why (1 3)(2 4) is even. A permutation is odd if it isn't even. (1 2 3 4 ) written as two cycles would be (1 4)(1 3)(1 2) which is odd.
 
  • #4
Thanks guys! I get it now :biggrin:
 

FAQ: Understanding Odd and Even Permutations in Linear Algebra - Quick Help

What are odd and even permutations in linear algebra?

In linear algebra, a permutation is a rearrangement of a set of elements. An odd permutation is one that requires an odd number of swaps to be sorted, while an even permutation requires an even number of swaps. In other words, an odd permutation changes the parity (even or odd) of the original set, while an even permutation does not.

How do I determine if a permutation is odd or even?

To determine the parity of a permutation, you can use the rule of alternation. Count the number of inversions (pairs of elements that are out of order) in the permutation. If the number of inversions is even, the permutation is even. If the number of inversions is odd, the permutation is odd.

What is the significance of odd and even permutations in linear algebra?

Odd and even permutations are important in understanding the structure of groups, which are sets of elements that can be combined in certain ways to produce new elements. In linear algebra, groups are often used to study matrices and their properties. Odd and even permutations help classify and define these groups.

Can a permutation be both odd and even?

No, a permutation cannot be both odd and even. The terms "odd" and "even" refer to the parity of the permutation, so a permutation can only be one or the other.

How are odd and even permutations used in practical applications?

Odd and even permutations have practical applications in fields such as computer science, cryptography, and coding theory. They are used to generate random numbers, create secure encryption algorithms, and analyze data structures. In linear algebra, they are used to study the properties and behavior of matrices, which have a wide range of applications in science, engineering, and technology.

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