- #1
Eclair_de_XII
- 1,083
- 91
- Homework Statement
- "The set ##\mathbb{C}## of complex numbers can be canonically identified with the space ##\mathbb{R}^2## by treating each ##z=x+iy\in \mathbb{C}## as a column ##(x,y)^T\in \mathbb{R}^2##.
(a) "Treating ##\mathbb{C}## as a complex vector space, show that multiplication by ##\alpha=a+ib\in \mathbb{C}## is a linear transformation in ##\mathbb{C}##. What is the matrix?"
- Relevant Equations
- ##(x+iy)(a+ib)=(xa-by)+i(ya+bx)##
If I had to guess what the complex matrix would look like, it would be:
##T(x+iy)=(xa-by)+i(ya+bx)=\begin{pmatrix}
a+bi & 0 \\
0 & -b+ai\end{pmatrix}\begin{pmatrix}
x \\
y \end{pmatrix}##
I'm not too sure on where everything goes; it's my first time fiddling with complex numbers, and moreover, using them in linear algebra.
##T(x+iy)=(xa-by)+i(ya+bx)=\begin{pmatrix}
a+bi & 0 \\
0 & -b+ai\end{pmatrix}\begin{pmatrix}
x \\
y \end{pmatrix}##
I'm not too sure on where everything goes; it's my first time fiddling with complex numbers, and moreover, using them in linear algebra.