Kunze Linear Algebra · Problem 17

Linear Transformations

Mathematics — Methods · commutator-subspace-trace-zero

Problem

Let WW be the space of n×nn \times n matrices over the field FF, and let W0W_0 be the subspace spanned by the matrices CC of the form C=AB−BAC = AB - BA. Prove that W0W_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−BAAB - BA.)
Intuitive explanation

The key insight is that commutators AB−BAAB - BA are trace-annihilating—they always have trace zero—so W0W_0 lives inside the trace-zero subspace. To prove equality, you need to show W0W_0 is large enough: since the trace-zero matrices form a codimension-1 subspace (dimension n2−1n^2 - 1), you just need to construct n2−1n^2 - 1 linearly independent commutators. The hint points you toward using elementary matrices (the "units" with a single 1 in position (i,j)(i,j)) to build enough commutators systematically to span the whole space.

Intuitive explanation

The key insight is that commutators AB−BAAB - BA are trace-annihilating—they always have trace zero—so W0W_0 lives inside the trace-zero subspace. To prove equality, you need to show W0W_0 is large enough: since the trace-zero matrices form a codimension-1 subspace (dimension n2−1n^2 - 1), you just need to construct n2−1n^2 - 1 linearly independent commutators. The hint points you toward using elementary matrices (the "units" with a single 1 in position (i,j)(i,j)) to build enough commutators systematically to span the whole space.

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