16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 17

Linear Transformations

Mathematics — Linear Algebra · commutator-subspace-trace-zero

Problem

Let

W

be the space of

n \times n

matrices over the field

F

, and let

W_0

be the subspace spanned by the matrices

C

of the form

C = AB - BA

. Prove that

W_0

is exactly the subspace of matrices which have trace zero. (Hint: What is the dimension of the space of matrices of trace zero? Use the matrix `units,' i.e., matrices with exactly one non-zero entry, to construct enough linearly independent matrices of the form

AB - BA

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