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Let T be the set of all matrics of the form AB - BA, where A and B are nxn matrics. Show that span T is not Mnn.


1) does "span T is not Mnn" mean that Mnn does not span T?




No it means that the set of all linear combinations of elements of T does not include all of Mnn.


One presumes that by Mnn you mean all nxn matrices, so the problem is to show that the space spanned by commutators is not all linear operators on n-space.


To do that you might want to look for an invariant shared by commutators that is not a property of all operators.

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