The following is a proof from Appendix C (Linear Spaces Review) of Introduction to Laplace Transforms and Fourier Series, Second Edition, by Phil Dyke:

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Should the highlighted part be $v_i$ instead of $u_i$? I don't understand how it could otherwise make sense.

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    $\begingroup$ Even worse, there is a typo in the headline: It should be "Gram-Schmidt"! $\endgroup$ – gerw Jul 3 '18 at 18:57
  • $\begingroup$ @gerw I suggest my students to write just GS in their handouts; I have a collection of about twenty different ways of writing it. In one case the student was of German mother language and she wrote wrongly “Schmidt”. When I asked her, she blushed and told me that her mother's maiden name is Schmidt! Not to mention in how many ways I've seen written Cauchy-Schwarz! $\endgroup$ – egreg Jul 3 '18 at 21:43

It's written correctly. By "the coefficient of $u_j$ in $u_i$" the author means the coefficient attached to $u_j$ in the equation defining $u_i$; e.g., the equation defining $u_3$ (which appears in the section you pasted into the question) is $u_3 = v_3 -c_2u_2 - c_1u_1$, where $c_2$ is the "coefficient of $u_2$ in $u_3$."

  • $\begingroup$ Ahh, I understand now. Thanks for the clarification. $\endgroup$ – The Pointer Jul 3 '18 at 17:35

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