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!
complex-analysis
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." â Scientifica, Jendrik Stelzner, Brahadeesh, Gibbs, amWhy
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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!
complex-analysis
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." â Scientifica, Jendrik Stelzner, Brahadeesh, Gibbs, amWhy
And here $T$ is ... ?
â MPW
Aug 25 at 11:27
T: C --> C is a R-linear map
â Jeez
Aug 25 at 11:28
2
Hello there. For an efficient interaction, please take a few minutes to check MathJax tutorial
â xbh
Aug 25 at 11:36
add a comment |Â
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!
complex-analysis
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!
complex-analysis
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." â Scientifica, Jendrik Stelzner, Brahadeesh, Gibbs, amWhy
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." â Scientifica, Jendrik Stelzner, Brahadeesh, Gibbs, amWhy
And here $T$ is ... ?
â MPW
Aug 25 at 11:27
T: C --> C is a R-linear map
â Jeez
Aug 25 at 11:28
2
Hello there. For an efficient interaction, please take a few minutes to check MathJax tutorial
â xbh
Aug 25 at 11:36
add a comment |Â
And here $T$ is ... ?
â MPW
Aug 25 at 11:27
T: C --> C is a R-linear map
â Jeez
Aug 25 at 11:28
2
Hello there. For an efficient interaction, please take a few minutes to check MathJax tutorial
â xbh
Aug 25 at 11:36
And here $T$ is ... ?
â MPW
Aug 25 at 11:27
And here $T$ is ... ?
â MPW
Aug 25 at 11:27
T: C --> C is a R-linear map
â Jeez
Aug 25 at 11:28
T: C --> C is a R-linear map
â Jeez
Aug 25 at 11:28
2
2
Hello there. For an efficient interaction, please take a few minutes to check MathJax tutorial
â xbh
Aug 25 at 11:36
Hello there. For an efficient interaction, please take a few minutes to check MathJax tutorial
â xbh
Aug 25 at 11:36
add a comment |Â
1 Answer
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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.
Yes, this is an answer to "I have no idea how to start the proof".
â GEdgar
Aug 25 at 13:31
add a comment |Â
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.
Yes, this is an answer to "I have no idea how to start the proof".
â GEdgar
Aug 25 at 13:31
add a comment |Â
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.
Yes, this is an answer to "I have no idea how to start the proof".
â GEdgar
Aug 25 at 13:31
add a comment |Â
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.
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.
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".
â GEdgar
Aug 25 at 13:31
add a comment |Â
Yes, this is an answer to "I have no idea how to start the proof".
â GEdgar
Aug 25 at 13:31
Yes, this is an answer to "I have no idea how to start the proof".
â GEdgar
Aug 25 at 13:31
Yes, this is an answer to "I have no idea how to start the proof".
â GEdgar
Aug 25 at 13:31
add a comment |Â
And here $T$ is ... ?
â MPW
Aug 25 at 11:27
T: C --> C is a R-linear map
â Jeez
Aug 25 at 11:28
2
Hello there. For an efficient interaction, please take a few minutes to check MathJax tutorial
â xbh
Aug 25 at 11:36