I am working through some problems in Axler's Linear Algebra Done Right textbook, and I noticed that I haven't really developed an intuitive feel for how to approach existence in the proofs.

The idea of existence just seems very vague to me. For instance, in one of the problems, the book asks you to find real numbers c and d given real numbers a and b in the following expression:

1/(a+bi) = c+di

From what it looks like, if I can express c and d in terms of a and b, that means that c and d exist because a and b are assumed to exist? Is there are more solid way for me to approach existence in proofs?

  • $\begingroup$ Note that $$\frac1{a+bi}=\frac{a-bi}{(a+bi)(a-bi)} = \frac{a-bi}{a^2+b^2}=\frac a{a^2+b^2}-i\frac b{a^2+b^2}$$So, if $c=\frac a{a^2+b^2},\;d=-\frac b{a^2+b^2}$ then $$\frac1{a+bi}=c+di$$ $\endgroup$ – Don Thousand May 14 at 2:17
  • $\begingroup$ I get this part, but I mean is it proof that something exists if it can be expressed in variables that we assume to exist already? $\endgroup$ – Richard K Yu May 14 at 2:24
  • $\begingroup$ I don't understand what you mean by "assume to exist already" The idea of existence is that given some complex number expressed generally as $a+bi$, if I can come up with a construction for the inverse also in the form $c+di$, then I've shown that one exists for all complex numbers. Not all existence proofs are constructive, but many are. $\endgroup$ – Don Thousand May 14 at 2:24
  • $\begingroup$ Welcome to Math Stack Exchange. If you can construct $c$ and $d$ using mathematical operations valid for real numbers (e.g., not dividing by zero, not taking a square root of a negative number), that is a solid proof of existence. In some situations (not this one) you might have only a non-constructive proof, which proves something exists without showing how to compute it $\endgroup$ – J. W. Tanner May 14 at 2:26
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    $\begingroup$ I'd suggest you read this wikipedia article $\endgroup$ – J. W. Tanner May 14 at 2:35

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