Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $w = a + bi$ be a complex number and let $T : \mathbb C\to \mathbb C$ be defined by $T (z) = w \cdot z$. Considering $\mathbb C$ as a vector space over $\mathbb R$, find the matrix $B$ representing $T$ relative to the basis $\{1, i\}$ of $\mathbb C$.

share|cite|improve this question
What did you try? – Sigur Dec 8 '12 at 19:36
up vote 4 down vote accepted

Applying $T$ on the basis:

$$T(1)=w=a\cdot 1+b\cdot i=(a,b)$$

and $$T(i)=w\cdot i=(a+bi)\cdot i=a\cdot i -b\cdot 1=(-b,a).$$

So $$T=\begin{pmatrix}a & -b \\ b &a\end{pmatrix}.$$

share|cite|improve this answer
Note that the matrices of this form provide a concrete construction of the complex numbers. – lhf Dec 8 '12 at 19:46

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.