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 $\mathbb C$.
Linear Algebra - complex numbers
1
$\begingroup$
linear-algebra
-
2What did you try? – 2012-12-08
1 Answers
5
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 $T=\begin{pmatrix}a & -b \\ b &a\end{pmatrix}.$
-
0Note that the matrices of this form provide a concrete construction of the complex numbers. – 2012-12-08