Every $mathbbR$-linear map $T colon mathbbC to mathbbC$ is of the form $T(z) = lambda z + mu overlinez$ [closed]

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Let $T colon mathbbC to mathbbC$ be an $mathbbR$-linear map.
Show that there exists complex numbers $lambda, mu$ such that one has $T(z) = lambda z + mu overlinez$ and show that $lambda, mu$ are uniquely determined by $T$, by giving explicit expressions of $lambda, mu$ in terms of $T(1)$ and $T(i)$.




Have no idea how to start the proof! Pls help!







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closed as off-topic by Scientifica, Jendrik Stelzner, Brahadeesh, Gibbs, amWhy Aug 26 at 18:15


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Scientifica, Jendrik Stelzner, Brahadeesh, Gibbs, amWhy
If this question can be reworded to fit the rules in the help center, please edit the question.












  • And here $T$ is ... ?
    – MPW
    Aug 25 at 11:27










  • T: C --> C is a R-linear map
    – Jeez
    Aug 25 at 11:28






  • 2




    Hello there. For an efficient interaction, please take a few minutes to check MathJax tutorial
    – xbh
    Aug 25 at 11:36















up vote
0
down vote

favorite













Let $T colon mathbbC to mathbbC$ be an $mathbbR$-linear map.
Show that there exists complex numbers $lambda, mu$ such that one has $T(z) = lambda z + mu overlinez$ and show that $lambda, mu$ are uniquely determined by $T$, by giving explicit expressions of $lambda, mu$ in terms of $T(1)$ and $T(i)$.




Have no idea how to start the proof! Pls help!







share|cite|improve this question














closed as off-topic by Scientifica, Jendrik Stelzner, Brahadeesh, Gibbs, amWhy Aug 26 at 18:15


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Scientifica, Jendrik Stelzner, Brahadeesh, Gibbs, amWhy
If this question can be reworded to fit the rules in the help center, please edit the question.












  • And here $T$ is ... ?
    – MPW
    Aug 25 at 11:27










  • T: C --> C is a R-linear map
    – Jeez
    Aug 25 at 11:28






  • 2




    Hello there. For an efficient interaction, please take a few minutes to check MathJax tutorial
    – xbh
    Aug 25 at 11:36













up vote
0
down vote

favorite









up vote
0
down vote

favorite












Let $T colon mathbbC to mathbbC$ be an $mathbbR$-linear map.
Show that there exists complex numbers $lambda, mu$ such that one has $T(z) = lambda z + mu overlinez$ and show that $lambda, mu$ are uniquely determined by $T$, by giving explicit expressions of $lambda, mu$ in terms of $T(1)$ and $T(i)$.




Have no idea how to start the proof! Pls help!







share|cite|improve this question















Let $T colon mathbbC to mathbbC$ be an $mathbbR$-linear map.
Show that there exists complex numbers $lambda, mu$ such that one has $T(z) = lambda z + mu overlinez$ and show that $lambda, mu$ are uniquely determined by $T$, by giving explicit expressions of $lambda, mu$ in terms of $T(1)$ and $T(i)$.




Have no idea how to start the proof! Pls help!









share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Aug 25 at 13:21









Jendrik Stelzner

7,57221037




7,57221037










asked Aug 25 at 11:25









Jeez

375




375




closed as off-topic by Scientifica, Jendrik Stelzner, Brahadeesh, Gibbs, amWhy Aug 26 at 18:15


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Scientifica, Jendrik Stelzner, Brahadeesh, Gibbs, amWhy
If this question can be reworded to fit the rules in the help center, please edit the question.




closed as off-topic by Scientifica, Jendrik Stelzner, Brahadeesh, Gibbs, amWhy Aug 26 at 18:15


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Scientifica, Jendrik Stelzner, Brahadeesh, Gibbs, amWhy
If this question can be reworded to fit the rules in the help center, please edit the question.











  • And here $T$ is ... ?
    – MPW
    Aug 25 at 11:27










  • T: C --> C is a R-linear map
    – Jeez
    Aug 25 at 11:28






  • 2




    Hello there. For an efficient interaction, please take a few minutes to check MathJax tutorial
    – xbh
    Aug 25 at 11:36

















  • And here $T$ is ... ?
    – MPW
    Aug 25 at 11:27










  • T: C --> C is a R-linear map
    – Jeez
    Aug 25 at 11:28






  • 2




    Hello there. For an efficient interaction, please take a few minutes to check MathJax tutorial
    – xbh
    Aug 25 at 11:36
















And here $T$ is ... ?
– MPW
Aug 25 at 11:27




And here $T$ is ... ?
– MPW
Aug 25 at 11:27












T: C --> C is a R-linear map
– Jeez
Aug 25 at 11:28




T: C --> C is a R-linear map
– Jeez
Aug 25 at 11:28




2




2




Hello there. For an efficient interaction, please take a few minutes to check MathJax tutorial
– xbh
Aug 25 at 11:36





Hello there. For an efficient interaction, please take a few minutes to check MathJax tutorial
– xbh
Aug 25 at 11:36











1 Answer
1






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1
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Hint: write $z=a+bi$ with $a,binBbb R$. Then by $Bbb R$-linearity, we have $T(z)=acdot T(1)+bcdot T(i)$.

Name the real and imaginary parts of $T(1)$ and $T(i)$ then continue the calculations.






share|cite|improve this answer




















  • Yes, this is an answer to "I have no idea how to start the proof".
    – GEdgar
    Aug 25 at 13:31

















1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
1
down vote













Hint: write $z=a+bi$ with $a,binBbb R$. Then by $Bbb R$-linearity, we have $T(z)=acdot T(1)+bcdot T(i)$.

Name the real and imaginary parts of $T(1)$ and $T(i)$ then continue the calculations.






share|cite|improve this answer




















  • Yes, this is an answer to "I have no idea how to start the proof".
    – GEdgar
    Aug 25 at 13:31














up vote
1
down vote













Hint: write $z=a+bi$ with $a,binBbb R$. Then by $Bbb R$-linearity, we have $T(z)=acdot T(1)+bcdot T(i)$.

Name the real and imaginary parts of $T(1)$ and $T(i)$ then continue the calculations.






share|cite|improve this answer




















  • Yes, this is an answer to "I have no idea how to start the proof".
    – GEdgar
    Aug 25 at 13:31












up vote
1
down vote










up vote
1
down vote









Hint: write $z=a+bi$ with $a,binBbb R$. Then by $Bbb R$-linearity, we have $T(z)=acdot T(1)+bcdot T(i)$.

Name the real and imaginary parts of $T(1)$ and $T(i)$ then continue the calculations.






share|cite|improve this answer












Hint: write $z=a+bi$ with $a,binBbb R$. Then by $Bbb R$-linearity, we have $T(z)=acdot T(1)+bcdot T(i)$.

Name the real and imaginary parts of $T(1)$ and $T(i)$ then continue the calculations.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Aug 25 at 13:14









Berci

56.9k23570




56.9k23570











  • Yes, this is an answer to "I have no idea how to start the proof".
    – GEdgar
    Aug 25 at 13:31
















  • Yes, this is an answer to "I have no idea how to start the proof".
    – GEdgar
    Aug 25 at 13:31















Yes, this is an answer to "I have no idea how to start the proof".
– GEdgar
Aug 25 at 13:31




Yes, this is an answer to "I have no idea how to start the proof".
– GEdgar
Aug 25 at 13:31


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