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    #1
    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


    ===================================================================================
    *** 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?

    =======================================================================================



    =======================================================================================

    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.

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