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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 $\mathbb C$.

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What did you try? –  Sigur Dec 8 '12 at 19:36

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

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

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