Search the corpus

    Real-linear but not complex-linear maps on $\CC$

    E-HK-31-10

    Area
    Algebra
    Topics
    Linear Algebra, Linear Transformations
    Status
    draft

    Let \(V\) be the set of all complex numbers regarded as a vector space over the field of real numbers (usual operations). Find a function from \(V\) into \(V\) which is a linear transformation on the above vector space, but which is not a linear transformation on \(C^1\), i.e., which is not complex linear.

    Solution

    Take complex conjugation, \[ T(z)=\overline z. \] For real scalars \(a,b\) and complex numbers \(z,w\), \[ T(az+bw)=\overline{az+bw}=a\overline z+b\overline w=aT(z)+bT(w), \] so \(T\) is real-linear.

    It is not complex-linear, since \[ T(i\cdot1)=T(i)=-i, \] whereas complex linearity would require \[ iT(1)=i. \] Thus \(T\) is linear over \(\RR\) but not over \(\CC\).

    Source Collections