Dimension

Let W be vector subspace of 2*2 matrices over R such that W is spanned by {AB-BA such that A,B are 2*2 matrices over R},then:
1)prove that dim(W)>=3 by verifyind E12,E21,E11-E22 all belong to W.
2)Considering the trace function tr:M2(R)-->R ,prove that dim(W)=3.
3)Let f:M2(R)-->R be a linear map such that f(AB)=f(BA),then prove that f=c.tr,for some c in R.Hence prove that f=tr if and only if f(Id)=2.

any help or suggestions rare appreciated..thank you

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Originally Posted by math.dj

Let W be vector subspace of 2*2 matrices over R such that W is spanned by {AB-BA such that A,B are 2*2 matrices over R},then:
1)prove that dim(W)>=3 by verifyind E12,E21,E11-E22 all belong to W.
2)Considering the trace function tr:M2(R)-->R ,prove that dim(W)=3.
3)Let f:M2(R)-->R be a linear map such that f(AB)=f(BA),then prove that f=c.tr,for some c in R.Hence prove that f=tr if and only if f(Id)=2.

any help or suggestions rare appreciated..thank you

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Well, what have you done? What are E12, E21, E11 and E22? Have you shown that E12, E21, and E11- E22 are of the form "AB- BA" for some A, B? To do that, you might want a simpler description of the space. What is true about the symmetry of AB-BA for any A and B?

What is the trace of such a matrix?