Welcome to our community

Be a part of something great, join today!

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

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,891
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:



Tapp - Defn 2.3 & Proposition 2.4 ... .png



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 ... ...


Tapp - 1 - Chapter 2, Section 1 - PART 1 ... .png
Tapp - 2 - Chapter 2, Section 1 - PART 2 ... .png



\(\displaystyle R_B\) (actually \(\displaystyle R_A\)) is defined in the following text ...



Tapp - Defn 1.9 & Defn 1.10 ... .png



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: