Are These Matrices a Basis for M2x2(R)?

In summary, the question asked if those vectors form a basis, and it was not clear if each individual vector was a basis. The answer is that each matrix must be invertible in order for them to be bases.
  • #1
karnten07
213
0

Homework Statement


Determine whether the following 2 x 2 matrices form a basis of the vector space M2x2(R) of all (2x2)-matrices over R:

A1=
1 0
0 0

A2=
2 2
0 0

A3=
3 2
1 0

A4=
4 3
2 1

Homework Equations





The Attempt at a Solution



So for them to be bases, they must be a linearly independant set that spans the vector space. How can i go about showing this for each matrix?
 
Physics news on Phys.org
  • #2
The question asked if those vectors form a basis. The question didn't ask if each individual vector was a basis (such a question wouldn't even make sense).
 
  • #3
Hurkyl said:
The question asked if those vectors form a basis. The question didn't ask if each individual vector was a basis (such a question wouldn't even make sense).

Do i need to show that each matrix is invertible?
 
  • #4
I think you need to start with the definitions, to make sure you understand just what the question is asking.
 
  • #5
Hurkyl said:
I think you need to start with the definitions, to make sure you understand just what the question is asking.

Oh so a1...a4 must all be linealry independant?
 
  • #6
karnten07 said:
Oh so a1...a4 must all be linealry independant?
Yes, that is what you need to show.
 
  • #7
Hurkyl said:
Yes, that is what you need to show.

Thanks, library closes now and no internet at new flat :cry:
 
  • #8
Try to solve c1*A1+c2*A2+c3*A3+c4*A4=0. It's pretty easy. What are the solutions for c1...c4? What does that say about linear independence?
 
  • #9
Hi, If I have a question i post it here and you will respond (if you are on line)?
 
  • #10
Hi DIck can you help me with this problem..."find the determinant of the linear transformation T(M)=[1,2,2,3]M+M[1,2,2,3] from the space V of symmetric 2x2 matrices to V
 
  • #11
In the future, always post a new thread for a new problem, ok? Otherwise this will get buried and not enough people will see it. The answer to your question is not obvious to me, maybe somebody else knows.
 
  • #12
hi dick thank you for responding. i posted a new thread. if i have to log off how will i be able to find it again? a response will be in my email box? i don't know how this forum works are there instructions somewhere?
 

1. What is the basis of a vector space?

The basis of a vector space is a set of linearly independent vectors that span the entire vector space. In other words, any vector in the space can be written as a unique linear combination of the basis vectors.

2. Why is the concept of a basis important in linear algebra?

The concept of a basis is important because it provides a way to represent and manipulate vectors in a vector space. It also allows for the simplification of complex linear systems and makes it easier to find solutions to linear equations.

3. How do you determine if a set of vectors form a basis?

To determine if a set of vectors form a basis, you need to check if the vectors are linearly independent and if they span the entire vector space. This can be done by setting up a system of equations and solving for the coefficients of the linear combination of the vectors. If the only solution is the trivial solution (all coefficients are zero), then the vectors are linearly independent and form a basis.

4. Can a vector space have more than one basis?

Yes, a vector space can have more than one basis. As long as the basis vectors are linearly independent and span the entire vector space, they can be considered as a valid basis. However, all bases for the same vector space will have the same number of vectors, known as the dimension of the vector space.

5. How is the basis of a vector space related to the concept of dimension?

The basis of a vector space is closely related to the dimension of the vector space. The dimension of a vector space is equal to the number of vectors in any basis for that vector space. In other words, the dimension tells us the minimum number of vectors needed to span the entire vector space.

Similar threads

  • Calculus and Beyond Homework Help
Replies
14
Views
531
  • Calculus and Beyond Homework Help
Replies
8
Views
624
  • Calculus and Beyond Homework Help
Replies
0
Views
418
  • Calculus and Beyond Homework Help
Replies
18
Views
1K
  • Calculus and Beyond Homework Help
Replies
15
Views
2K
  • Calculus and Beyond Homework Help
Replies
2
Views
927
  • Calculus and Beyond Homework Help
2
Replies
58
Views
3K
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
7
Views
2K
  • Calculus and Beyond Homework Help
Replies
4
Views
2K
Back
Top