Are These Matrices a Basis for M2x2(R)?

[karnten07]

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 independent set that spans the vector space. How can i go about showing this for each matrix?

 

[Hurkyl]

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).

 

[karnten07]

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?

 

[Hurkyl]

I think you need to start with the definitions, to make sure you understand just what the question is asking.

 

[karnten07]

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 independent?

 

[Hurkyl]

Oh so a1...a4 must all be linealry independent?

Yes, that is what you need to show.

 

[karnten07]

Yes, that is what you need to show.

Thanks, library closes now and no internet at new flat

 

[Dick]

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?

 

[riordo]

Hi, If I have a question i post it here and you will respond (if you are on line)?

 

[riordo]

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

 

[Dick]

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.

 

[riordo]

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?