Linear Algebra/Complex Representations

Recall the definitions of the complex number addition



(a+bi)\,+\,(c+di)=(a+c)+(b+d)i $$

and multiplication.


 * $$\begin{array}{rl}

(a+bi)(c+di) &=ac+adi+bci+bd(-1) \\ &=(ac-bd)+(ad+bc)i \end{array}$$

Handling scalar operations with those rules, all of the operations that we've covered for real vector spaces carry over unchanged.

Everything else from prior chapters that we can, we shall also carry over unchanged. For instance, we shall call the ordered set of vectors



\left\langle \begin{pmatrix} 1+0i \\ 0+0i \\ \vdots \\ 0+0i \end{pmatrix}, \begin{pmatrix} 0+0i \\ 1+0i \\ \vdots \\ 0+0i \end{pmatrix}, \dots, \begin{pmatrix} 0+0i \\ 0+0i \\ \vdots \\ 1+0i \end{pmatrix} \right\rangle $$

the standard basis for $$ \mathbb{C}^n $$ as a vector space over $$\mathbb{C}$$ and again denote it $$\mathcal{E}_n $$.