In Hoffman & Kunze Linear Algebra book (Ch.1 Page 4), there is a saying that "if we have another system of linear equations in which each of the k equations is a linear combination of the equations from the main system, then every solution of the main system is a solution for the second system" (until that point I understand).

Then it says: "Of course it may happen that some solution of the second system are not solutions of the first system".

I can't think of an example, why is it true?

  • $\begingroup$ Suppose the original set of equations is independent, but the derived system has some dependent equations. $\endgroup$ – amd Oct 27 '17 at 20:14
  • $\begingroup$ Can you provide an actual example? $\endgroup$ – moshek Oct 28 '17 at 18:51
  • $\begingroup$ $x=0, y=0$ has exactly one solution; $x+y=0, 2x+2y=0$ has an infinite number of them. $\endgroup$ – amd Oct 28 '17 at 18:58
  • $\begingroup$ Thanks, solved my problem, you can post it as an answer. $\endgroup$ – moshek Oct 28 '17 at 19:04

In forming these linear combinations, you might introduce dependencies that didn’t exist in the original system. A simple example: The system $x=0$, $y=0$ has a single solution; the system $x+y=0$, $2x+2y=0$ has an infinite number of them, including the solution of the original system.

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  • $\begingroup$ In linear-algebraic terms, you might have introduced new dependencies so that the null space of the new system’s coefficient matrix is a superset of the null space of the original system’s matrix. $\endgroup$ – amd Oct 28 '17 at 19:10

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