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Could you please explain me more step by step the solution to problem 5 in the link? I cannot understand the logic flow behind it. For example, why for $(a1, a2,\dots, a_m)$ there is $a_i \ne 0$ and not $a_1 = 1$ as stated in the problem?

Thank you for help.

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closed as unclear what you're asking by Brian Fitzpatrick, Daniel, Harish Chandra Rajpoot, msteve, hardmath Aug 11 '15 at 4:11

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ A quick statement that may help: if you have a linear operator $T : V \to W$ and have bases for $V$ and $W$, then $T_{ij}$ is the coefficient of $w_i$ in the expansion of $Tv_j$ in the basis for $W$. $\endgroup$ – Ian Aug 10 '15 at 20:28
  • $\begingroup$ The body of your Question should be as self-contained as possible. Please restate the problem 5 and the outlined solution in your own words, as this will help the Reader understand what assistance you want. $\endgroup$ – hardmath Aug 11 '15 at 4:11
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You want to show that $a_1$ can be made to either one or zero be picking the right base, besides other stuff. If all $a_i=0$, you are done. If there is one $a_i\neq 0$, you can use the base transformation shown in the proof to create a new base with $(a_1,a_2,a_3,...) = (1,0,0,...)$.

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