Complex-Linear Matrices & C-Linear Transformations ... Tapp, Propn 2.4 ...

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Summary:

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.

Main Question or Discussion Point

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 (##F##) is ##\mathbb{C}##-linear if and only if ##F(i \cdot X) = i \cdot F(X)## for all ##X \in \mathbb{C}^n## ... "


My question is as follows ... can someone please demonstrate a proof of the fact that ##F## is ##\mathbb(C)##-linear if and only if ##F(i \cdot X) = i \cdot F(X)## for all ##X \in \mathbb{C}^n## ...



Help will be much appreciated ...

Peter


===================================================================================
*** EDIT ***

After a little reflection it appears that " ... ##F## is ##\mathbb{C}##-linear ##\Longrightarrow F(i \cdot X) = i \cdot F(X)## ... " is immediate as ...

... taking ##c = i## we have ...

##F(c \cdot X ) = c \cdot F(X) \Longrightarrow F(i \cdot X) = i \cdot F(X)## for ##c \in \mathbb{C}##


Is that correct?

Peter

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



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

Note that Tapp defines##\rho_n## and ##f_n## in the following text ... ...



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

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




Also note that ##R_B## (actually ##R_A##) is defined in the following text ...




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



Hope the provision of the above text helps with definitions, notation and context ...

Peter
 

Answers and Replies

  • #2
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The proof uses that a map ##T:V\to W## between ##\mathbb{C}##-vector spaces is ##\mathbb{C}##-linear if and only if if ##\mathbb{R}##-linear and ##T(iv)=iT(v)## for all ##v\in V##.

From left to right is obvious by the definition, from right to left requires a little easy two-line proof.

Note that in the proof one says that ##T## is ##\mathbb{C}## linear if and only if ##T(iv) = iT(v)## for all ##v\in V## since it was already established that ##T## was ##\mathbb{R}##-linear.
 
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  • #3
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The proof uses that a map ##T:V\to W## between ##\mathbb{C}##-vector spaces is ##\mathbb{C}##-linear if and only if if ##\mathbb{R}##-linear and ##T(iv)=iT(v)## for all ##v\in V##.

From left to right is obvious by the definition, from right to left requires a little easy two-line proof.

Note that in the proof one says that ##T## is ##\mathbb{C}## linear if and only if ##T(iv) = iT(v)## for all ##v\in V## since it was already established that ##T## was ##\mathbb{R}##-linear.

Thanks for the post and the hint ...

We are considering ##F \ : \ \mathbb{C}^n \to \mathbb{C}^n## where ##F = f_n^{ -1 } \circ R_B \circ f_n## ...

We know that ##F## is ##\mathbb{R}##-linear ... that is ...

##F(X + Y) = F(X) + F(Y)## for all ##X, Y \in \mathbb{C}^n## ... ... ... ... ... (1)

... and ...

##F(rX) = r F(X)## for all ##X \in \mathbb{C}^n## and ##r \in \mathbb{R}## ... ... ... ... ... (2)


We have to prove that ##F## is ##\mathbb{C}##-linear ... ...

... that is ...

##F(X + Y) = F(X) + F(Y)## for all ##X, Y \in \mathbb{C}^n## ...

... which holds from the ##\mathbb{R}##-linear case ...

and ...

##F(cX) = cF(X)## for all ##X \in \mathbb{C}^n## and ##c \in \mathbb{C}## ... ... ... ... ... (3)


So ... essentially we have to prove (3) ...

So then ... let ##c = a + bi## ... and proceed as follows ...

##F(cX) = F( (a+bi)X)##

##= F(aX + biX)##

##= F(aX) + F(biX)## since ##F## is ##\mathbb{R}##-linear

##= F(aX) + iF(bX)## since ##F(iY) = iF(Y)##

##= aF(X) + biF(X)## since ##F## is ##\mathbb{R}##-linear

##= (a +bi)F(X)##

##= cF(X)##


Hope the above is correct ... ...

Peter
 
  • #4
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Thanks for the post and the hint ...

We are considering ##F \ : \ \mathbb{C}^n \to \mathbb{C}^n## where ##F = f_n^{ -1 } \circ R_B \circ f_n## ...

We know that ##F## is ##\mathbb{R}##-linear ... that is ...

##F(X + Y) = F(X) + F(Y)## for all ##X, Y \in \mathbb{C}^n## ... ... ... ... ... (1)

... and ...

##F(rX) = r F(X)## for all ##X \in \mathbb{C}^n## and ##r \in \mathbb{R}## ... ... ... ... ... (2)


We have to prove that ##F## is ##\mathbb{C}##-linear ... ...

... that is ...

##F(X + Y) = F(X) + F(Y)## for all ##X, Y \in \mathbb{C}^n## ...

... which holds from the ##\mathbb{R}##-linear case ...

and ...

##F(cX) = cF(X)## for all ##X \in \mathbb{C}^n## and ##c \in \mathbb{C}## ... ... ... ... ... (3)


So ... essentially we have to prove (3) ...

So then ... let ##c = a + bi## ... and proceed as follows ...

##F(cX) = F( (a+bi)X)##

##= F(aX + biX)##

##= F(aX) + F(biX)## since ##F## is ##\mathbb{R}##-linear

##= F(aX) + iF(bX)## since ##F(iY) = iF(Y)##

##= aF(X) + biF(X)## since ##F## is ##\mathbb{R}##-linear

##= (a +bi)F(X)##

##= cF(X)##


Hope the above is correct ... ...

Peter
Yes, completely correct. Well done!
 
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