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I am trying to understand the proof, but am stuck at the last line. Why does the first equality of the picture below holds? The codomain of $f$ is $R^k$, not some the set of nonnegative numbers. Is there something I am missing? I need hekp.

enter image description here

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  • $\begingroup$ It might be a closure of a set. $\endgroup$ – Atbey Jan 20 '18 at 9:16
  • $\begingroup$ What do you mean by "it"? Could you explain in more detail? $\endgroup$ – Keith Jan 20 '18 at 9:18
  • $\begingroup$ Why did you mention nonnegative numbers? $\endgroup$ – Atbey Jan 20 '18 at 9:27
  • $\begingroup$ Because, nonzero elements in $R^k$ can add up to zero. So, there is no guarantee that the inclusion of the right side into left side on the first equality holds. $\endgroup$ – Keith Jan 20 '18 at 9:36
  • $\begingroup$ You should mention the reference. $\endgroup$ – Sou Jan 20 '18 at 18:07

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