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    The commutator map $A \mapsto AB - BA$ is linear

    E-HK-31-9

    Area
    Algebra
    Topics
    Linear Algebra, Linear Transformations
    Status
    draft

    Let \(V\) be the vector space of all \(n \times n\) matrices over the field \(F\), and let \(B\) be a fixed \(n \times n\) matrix. If \[ T(A) = AB - BA \] verify that \(T\) is a linear transformation from \(V\) into \(V\).

    Solution

    For matrices \(A,C\in V\) and scalars \(r,s\in F\), \[ T(rA+sC)=(rA+sC)B-B(rA+sC). \] Distributing matrix multiplication gives \[ T(rA+sC)=r(AB-BA)+s(CB-BC)=rT(A)+sT(C). \] Thus \(T\) is linear from \(V\) to \(V\).

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