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I am working through the book Linear Algebra Done Right. I am stumped on this exercise question.

I used ChatGPT and understand up to here:

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I just don't understand how I deduce that the real part=0 and the imaginary=1? Is this a fundamental concept I'm unaware of? Something else?

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  • $\begingroup$ Do you understand what "Equate real and imaginary parts" means? $\endgroup$ Commented Jun 21 at 13:04
  • $\begingroup$ I just figured it out. Since the statement is set to i, a^2-b^2 must be 0, while 2ab must be 1. $\endgroup$ Commented Jun 21 at 13:08
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    $\begingroup$ This is the first time I have seen ChatGPT successfully complete a proof or derivation with no errors. Well done to it, I guess... $\endgroup$ Commented Jun 21 at 13:24
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    $\begingroup$ Don't use ChatGPT for anything except personal amusement. It is not a search engine, it is not a valid reference. It is a synthetic text extruder that works by deciding what is the likely next word on the basis of what the previous words and prompt were. This is a classic example. You want two square roots of the identity linear transformation. It is giving you two complex square roots of the imaginary number $i$. $\endgroup$ Commented Jun 21 at 19:55
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    $\begingroup$ @ChristianE.Ramirez Except that it is extruding a proof to the wrong question. But what can you expect of a stochastic parrot? $\endgroup$ Commented Jun 21 at 19:57

2 Answers 2

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In the definition $z = a+bi, \; a, b \in \mathbb R $.

And since $\mathbb R+\mathbb R =\mathbb R$, means that there is no sum of real numbers that is an imaginary number this means that only $2abi$ can contribute to the imaginary part of $z$ and only $a^2-b^2$ to the real one. Notice that $i = 0+i$, the real part is zero: so we get a system of equations: $$ \cases{a^2-b^2 = 0\\2abi=i}\implies\cases{a^2-b^2=0\\b = \frac1{2a}} $$ Substituting $c = a^2$: $$ c-0.25c^{-1}=0\implies c^2 -0.25=0\implies c=\pm0.5\implies b=a = \pm\frac{\sqrt{2}}2$$ So $z = \pm\frac{\sqrt{2}}2(1+i)$

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$a$ and $b$ are real. So the complex number $a^2-b^2+2abi$ can be expressed as a purely real component $a^2-b^2$ plus a purely imaginary component $2abi$.

Also, the complex number $i$ can be expressed as a purely real component $0$ plus a purely imaginary component $i$.

But these two complex numbers are equal. Therefore $a^2-b^2=0$ and $ab=\frac12$.

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