11  The complex numbers as a real vector space

Identifying \(x + \mathrm{i}y\) with \((x,y)\) turns the complex plane into a two-dimensional real vector space.

The geometrical interpretation of the \(\mathbb{C}\) as the plane \(\mathbb{R}^2\) indicates that \(\mathbb{C}\) can be viewed as a two-dimensional real vector space. Indeed, addition and multiplication with real scalars are compatible with the axioms for Euclidean space \(\mathbb{R}^2\).

Exercises

Exercise 11.1 (Complex numbers as a real vector space)  

1. Show that \(\mathbb{C}\) regarded can be regarded as the Euclidean plane: Check axioms for Euclidean space and check that vector addition and multiplication with real scalars are the same in the two spaces. Check also that the Euclidean norm is the modulus of the complex number. What are the complex numbers that correspond to the standard basis in \(\mathbb{R}^2\)? Conclude that \(\mathbb{C}\) can be regarded as a real Euclidean vector space of dimension \(2\).

2. Show that multiplication with a complex number \(z\) is a linear operator on \(\mathbb{R}^2\), and find its matrix in the standard basis. What is the matrix of multiplication with \(\mathrm{i}\)?

3. Show that multiplication with a complex number of modulus 1 is a rotation in \(\mathbb{R}^2\).

4. Show that the map \(z \mapsto \bar{z}\) is a linear transformation in \(\mathbb{R}^2\). What kind of linear transformation is this? Is it a linear transformation in \(\mathbb{C}\)?