Kunze Linear Algebra · Problem 3.1.10
Linear Transformations
Mathematics — Methods · real-vs-complex-linear
Problem
Let be the set of all complex numbers regarded as a vector space over the field of real numbers (usual operations). Find a function from into which is a linear transformation on the above vector space, but which is not a linear transformation on , i.e., which is not complex linear.Intuitive explanation
The key is to exploit the difference between real and complex linearity. A real-linear map on can treat the real and imaginary parts independently, while complex-linearity requires respecting scalar multiplication by all complex numbers, not just real ones. Try a map that mixes real and imaginary parts in a way that works for real scaling but breaks under multiplication by .
Intuitive explanation
The key is to exploit the difference between real and complex linearity. A real-linear map on can treat the real and imaginary parts independently, while complex-linearity requires respecting scalar multiplication by all complex numbers, not just real ones. Try a map that mixes real and imaginary parts in a way that works for real scaling but breaks under multiplication by .