Connecting linear algebra concepts to groups

In summary: G.Good point! My gut feeling just told me that the determinant is the answer but my brain told me to be careful.
  • #1
lemonthree
51
0
question.png


The options are
\(\displaystyle rank(B)+null(B)=n\)
\(\displaystyle tr(ABA^{−1})=tr(B)\)
\(\displaystyle det(AB)=det(A)det(B)\)

I'm thinking that since it's invertible, I would focus on the determinant =/= 0. I believe the first option is out, because null (B) would be 0 which won't be helpful. The second option makes the point that \(\displaystyle AA^{−1}\) is \(\displaystyle I\), so it's suggesting invertibility. So I'm deciding between the second and last option. Does anyone have any tips?
 
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  • #2
Since the problem specifically refers to "the map defined by taking determinants", I would almost automatically check "det(AB)= det(A)det(B)"! "Almost" because, of course, I would want to make sure it was correct.
 
  • #3
Country Boy said:
Since the problem specifically refers to "the map defined by taking determinants", I would almost automatically check "det(AB)= det(A)det(B)"! "Almost" because, of course, I would want to make sure it was correct.

Agree, I was tempted to select det(AB)= det(A)det(B) but then again I can't be fully sure.

I guess since we both believe it to be correct, I guess I'm going for this answer then!
 
  • #4
I didn't say I believe it to be correct! I said I would "almost" automatically check it. And I said "of course, I would want to make sure it was correct". Why can't you "be fully sure"?
 
  • #5
Country Boy said:
I didn't say I believe it to be correct! I said I would "almost" automatically check it. And I said "of course, I would want to make sure it was correct". Why can't you "be fully sure"?

Good point! My gut feeling just told me that the determinant is the answer but my brain told me to be careful.
 
  • #6
Let's take a closer look, and let's start with the crucial bit of information that we need: what is the definition of a group homomorphism?
 
  • #7
Hi Klaas, indeed, it is det(AB) = det(A)det(B) because as per the definition of group homomorphism, φ (G) = H if φ (xy) = φ (x) φ (y)
 

1. How are linear algebra concepts related to groups?

Linear algebra and group theory are both branches of mathematics that deal with abstract structures and their properties. Groups can be defined as sets of elements that follow a specific set of rules, while linear algebra deals with vector spaces and the operations on them. The concept of a group can be applied to linear algebra by considering vector spaces as groups and studying their properties under different operations.

2. What are the main differences between linear algebra and group theory?

While both linear algebra and group theory deal with abstract structures, they have different focuses. Linear algebra is primarily concerned with vector spaces and their properties, while group theory is focused on the properties and structures of groups. Additionally, linear algebra uses operations such as addition and multiplication, while group theory uses operations such as composition and inversion.

3. How can understanding groups improve my understanding of linear algebra?

Studying groups can provide a deeper understanding of the properties and structures of vector spaces. By considering vector spaces as groups, one can apply the principles and theorems of group theory to linear algebra, leading to a more comprehensive understanding of the subject. This can also help in solving more complex problems and proofs in linear algebra.

4. What are some real-world applications of connecting linear algebra concepts to groups?

The application of group theory to linear algebra has numerous real-world applications, particularly in fields such as physics, computer science, and cryptography. For example, group theory is used in quantum mechanics to describe the symmetries of physical systems, and in cryptography to develop secure encryption algorithms.

5. Are there any resources available for learning about connecting linear algebra concepts to groups?

Yes, there are many resources available for learning about the connection between linear algebra and groups. These include textbooks, online courses, and lecture notes from universities. Additionally, there are many online forums and communities where one can discuss and learn about this topic from experts and other learners.

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