
MHB Master
#1
March 1st, 2020,
00:14
I am reading Kristopher Tapp's book: Matrix Groups for Undergraduates.
I am currently focused on and studying Section 1 in Chapter2, namely:
"1. Complex Matrices as Real Matrices".
I need help in fully understanding how to prove an assertion related to Tapp's Proposition 2.4.
Proposition 2.4 and some comments following it read as follows:
In the remarks following Proposition 2.4 we read the following:
" ... ... It ($ \displaystyle F$) is $ \displaystyle \mathbb{C}$linear if and only if $ \displaystyle F(i \cdot X) = i \cdot F(X)$ for all $ \displaystyle X \in \mathbb{C}^n$ ... "
My question is as follows ... can someone please demonstrate a proof of the fact that $ \displaystyle F$ is $ \displaystyle \mathbb(C)$linear if and only if $ \displaystyle F(i \cdot X) = i \cdot F(X)$ for all $ \displaystyle X \in \mathbb{C}^n$ ...
Note that even a indication of the main steps of the proof would help ...
Help will be much appreciated ...
Peter
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*** EDIT ***
After a little reflection it appears that " ... $ \displaystyle F$ is $ \displaystyle \mathbb{C}$linear $ \displaystyle \Longrightarrow$ $ \displaystyle F(i \cdot X) = i \cdot F(X)$ ... " is immediate as ...
... taking $ \displaystyle c = i$ we have ...
$ \displaystyle F(c \cdot X ) = c \cdot F(X)$ $ \displaystyle \Longrightarrow$ $ \displaystyle F(i \cdot X) = i \cdot F(X)$ for $ \displaystyle c \in \mathbb{C}$
Is that correct?
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Tapp defines $ \displaystyle \rho_n$ and $ \displaystyle f_n$ in the following text ... ...
$ \displaystyle R_B$ (actually $ \displaystyle R_A$) is defined in the following text ...
Note that Tapp uses $ \displaystyle \mathbb{K}$ to denote one of $ \displaystyle \mathbb{R}, \mathbb{C}$, or $ \displaystyle \mathbb{H}$ ... ...
Hope that helps ...
Peter
Last edited by Peter; March 2nd, 2020 at 00:54.

March 1st, 2020 00:14
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