The following is a proof from Appendix C (Linear Spaces Review) of Introduction to Laplace Transforms and Fourier Series, Second Edition, by Phil Dyke:
Should the highlighted part be $v_i$ instead of $u_i$? I don't understand how it could otherwise make sense.
Best Answer
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 $u_3$."