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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

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

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