Basis of a vector space

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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?
 

Answers and Replies

  • #2
Hurkyl
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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
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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
Hurkyl
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I think you need to start with the definitions, to make sure you understand just what the question is asking.
 
  • #5
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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
Hurkyl
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Oh so a1...a4 must all be linealry independant?
Yes, that is what you need to show.
 
  • #7
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Yes, that is what you need to show.

Thanks, library closes now and no internet at new flat :cry:
 
  • #8
Dick
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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
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Hi, If I have a question i post it here and you will respond (if you are on line)?
 
  • #10
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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
Dick
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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
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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?
 

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