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# Thread: Complex-Linear Matrices and C-Linear Transformations ... Tapp, Propostion 2.4 ... ...

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.

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

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