Homomorphism Help: Show det(A) is a Homomorphism to R*

  • Thread starter proplaya201
  • Start date
In summary, the conversation discusses the determinant mapping from the group of nonzero real numbers under multiplication to the group of invertible 2x2 matrices over the real numbers. It is shown that this mapping is a homomorphism with the kernel being the special linear group. The person involved is unsure how to proceed with proving this homomorphism.
  • #1
proplaya201
14
0

Homework Statement



let R* be the group of nonzero real numbers under multiplication. then the determinant mapping A-> det(A) is a homomorphism from GL(2,R) to R*. the kernel of the determinant mapping is SL(2,R).
i am suppose to show that this is a homomorphism but i have no idea where to go and what to do

Homework Equations


det(AB) = det(A)det(B)

The Attempt at a Solution

 
Physics news on Phys.org
  • #2
If you don't realize that det(AB)=det(A)det(B) basically proves that it is a homomorphism, you might want to look up the definition of homomorphism and think about it.
 

1. What is a homomorphism?

A homomorphism is a function that preserves the algebraic structure between two mathematical objects. In other words, it maps elements of one set to elements of another set in a way that respects the operations of the sets.

2. How can we show that det(A) is a homomorphism?

To show that det(A) is a homomorphism, we need to demonstrate that it preserves the operation of multiplication. This means that for two matrices A and B, det(A*B) = det(A) * det(B).

3. What does it mean for det(A) to be a homomorphism to R*?

Det(A) being a homomorphism to R* means that it maps the set of matrices to the set of non-zero real numbers, preserving the operation of multiplication. In other words, the determinant function takes a matrix and produces a non-zero real number as its output.

4. Why is it important to show that det(A) is a homomorphism to R*?

Showing that det(A) is a homomorphism to R* is important because it provides a way to simplify calculations involving determinants. By preserving the operation of multiplication, we can use the properties of homomorphisms to simplify the determinant of a product of matrices to the product of their determinants.

5. Can you provide an example of how to show det(A) is a homomorphism to R*?

Yes, let's consider two 2x2 matrices, A = [a b; c d] and B = [e f; g h]. To show that det(A) is a homomorphism, we need to demonstrate that det(A*B) = det(A) * det(B). This can be done by calculating the determinant of each matrix and showing that their product is equal. For example, det(A*B) = det([ae+bg af+bh; ce+dg cf+dh]) = (ae+bg)(cf+dh)-(af+bh)(ce+dg) = (ad-bc)(eh-fg) = det(A) * det(B).

Similar threads

  • Calculus and Beyond Homework Help
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
14
Views
2K
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
17
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
2
Replies
36
Views
2K
  • Calculus and Beyond Homework Help
Replies
9
Views
3K
  • Calculus and Beyond Homework Help
Replies
11
Views
1K
  • Calculus and Beyond Homework Help
Replies
7
Views
2K
  • Calculus and Beyond Homework Help
Replies
4
Views
4K
Back
Top