Vector spaces and spanning sets

  • Thread starter misterau
  • Start date
1. The problem statement, all variables and given/known data
Knowing this set spans M22:

[1 , 0] , [0 , 1] , [0 , 0] ,[0 , 0]
[0 , 0] , [0 , 0] , [1 , 0] ,[0 , 1]

What is another spanning set for this vector space? Justify your choice by showing that it is a linearly independent set.

3. The attempt at a solution

[2 , 0] , [0 , 2] , [0 , 0] ,[0 , 0]
[0 , 0] , [0 , 0] , [2 , 0] ,[0 , 2]

I am I on the right track?
 
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This question is not well worded, IMO. You could add any old 2x2 matrix to the original set and still have a spanning set, but the new set would not be linearly independent, which is not a requirement of a spanning set. This is, however, a requirement of a minimal spanning set, which is the same as a basis.

Your set of matrices is also a spanning set. To convince yourself of this show that any matrix M in M22 can be written as a linear combination of the elements in your set. I.e., c1*M1 + c2*M2 + c3*M3 + c4*M4 = M, where the Mi's are the matrices in your set.

To show that your set of matrices is linearly independent, show that the equation
c1*M1 + c2*M2 + c3*M3 + c4*M4 = 0 has exactly one solution: c1 = c2 = c3 = c4 = 0.
 
Thanks for the help!
 

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