Subspace Problems: Which R^n*n Subsets are Subspaces?

  • Thread starter dkyo
  • Start date
  • Tags
    Subspace
In summary, the conversation is discussing which subsets of R^n*n are subspaces of R^n*n. The suggested subsets include symmetric matrices, diagonal matrices, nonsingular matrices, singular matrices, triangular matrices, upper triangular matrices, matrices that commute with a given matrix A, matrices such that A^2 = A, and matrices such that trace(A) = 0. The conversation also mentions that no one will provide a detailed solution, but rather guide the individual towards finding their own solution.
  • #1
dkyo
1
0

Homework Statement



Which of the following subsets of R^n*n are in fact subspaces of R^n*n

1) The symmetric matrices
2) The diagonal matrices
3) The nonsingular matrices
4) The singular matrices
5) The triangular matrices
6) The upper triangular matrices
7) All matrices that commute with a given matrix A
8) All matices such that A^2 = A
9) All matrices such that trace(A) = 0

Can anyone give me a detailed solution for this questions?

Homework Equations





The Attempt at a Solution

 
Physics news on Phys.org
  • #2
First of all, what are the qualities a subset needs to have to be a subspace of R^n*n?
 
  • #3
No one will 'give you a detailed solution', they will only help you to arrive at your own solution. nicktacik has suggested how to start thinking about the problem. I suggest you start.
 

1. What is a subspace?

A subspace is a subset of a vector space that satisfies the three properties of closure under addition, closure under scalar multiplication, and contains the zero vector.

2. How do you determine if a subset is a subspace?

To determine if a subset is a subspace, you must check if it satisfies the three properties of closure under addition, closure under scalar multiplication, and contains the zero vector. If all three properties are satisfied, then the subset is a subspace.

3. Can a subspace have more than one dimension?

Yes, a subspace can have more than one dimension. A subspace is a subset of a vector space, so it inherits the dimension of the vector space it is a part of.

4. What are the benefits of studying subspace problems?

Studying subspace problems can help improve our understanding of vector spaces and linear algebra, which have many applications in fields such as physics, engineering, and computer science. It also helps us develop problem-solving and critical thinking skills.

5. Are there any real-world applications of subspace problems?

Yes, there are many real-world applications of subspace problems. For example, in computer graphics, subspace methods are used to create realistic animations and simulations. Subspaces are also used in data analysis and machine learning to reduce the dimensionality of large datasets and extract important features.

Similar threads

  • Calculus and Beyond Homework Help
Replies
8
Views
628
  • Calculus and Beyond Homework Help
Replies
15
Views
2K
  • Calculus and Beyond Homework Help
Replies
9
Views
2K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
5
Views
4K
  • Calculus and Beyond Homework Help
Replies
26
Views
2K
  • Calculus and Beyond Homework Help
Replies
15
Views
1K
  • Calculus and Beyond Homework Help
Replies
5
Views
4K
  • Calculus and Beyond Homework Help
Replies
7
Views
1K
  • Calculus and Beyond Homework Help
Replies
8
Views
1K
Back
Top