Recall the definitions of the complex number addition
![{\displaystyle (a+bi)\,+\,(c+di)=(a+c)+(b+d)i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3d9c25c4d035cb55fce7fe838c1285ae6a6afba)
and multiplication.
![{\displaystyle {\begin{array}{rl}(a+bi)(c+di)&=ac+adi+bci+bd(-1)\\&=(ac-bd)+(ad+bc)i\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/642b359a5883525aa4ef485c0c60f4edadf87da7)
- Example 2.1
For instance,
and
.
Handling scalar operations with those rules, all of
the operations that we've covered
for real vector spaces carry over unchanged.
- Example 2.2
Matrix multiplication is the same, although the scalar arithmetic involves more
bookkeeping.
![{\displaystyle {\begin{aligned}&={\begin{pmatrix}(1+1i)\cdot (1+0i)+(2-0i)\cdot (3i)&(1+1i)\cdot (1-0i)+(2-0i)\cdot (-i)\\(i)\cdot (1+0i)+(-2+3i)\cdot (3i)&(i)\cdot (1-0i)+(-2+3i)\cdot (-i)\end{pmatrix}}\\&={\begin{pmatrix}1+7i&1-1i\\-9-5i&3+3i\end{pmatrix}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0134b555d73963c8a25df83da1a589777958c232)
Everything else from prior chapters that we can,
we shall also carry over unchanged.
For instance, we shall call this
![{\displaystyle \langle {\begin{pmatrix}1+0i\\0+0i\\\vdots \\0+0i\end{pmatrix}},\dots ,{\begin{pmatrix}0+0i\\0+0i\\\vdots \\1+0i\end{pmatrix}},{\begin{pmatrix}0+1i\\0+0i\\\vdots \\0+0i\end{pmatrix}},\dots ,{\begin{pmatrix}0+0i\\0+0i\\\vdots \\0+1i\end{pmatrix}}\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/12d9fc600c94bdf3e8cb91401e7a8b0513f09697)
the standard basis for
as a vector
space over
and again denote it
.