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Consider "any" two systems of linear equations: Sys_A and Sys_B.

Are the below two statements equivalent?

Statement (1): Sys_A and Sys_B have same solution space

Statement (2): Sys_A and Sys_B are mutually linearly dependent

Statement (2) implies Statement (1) is rather obvious, I'm not sure about the other direction

Thanks

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  • $\begingroup$ What does it mean for two linear systems to be linearly dependent? $\endgroup$ – tst Aug 24 '18 at 10:27
  • $\begingroup$ That every equation of one system is a linear combination of the equations in the other system. Sorry if that was not made clear, Thanks $\endgroup$ – aman_cc Aug 24 '18 at 10:31
  • $\begingroup$ This implies that there is a change of basis that turns one system to the other. $\endgroup$ – tst Aug 24 '18 at 10:34
  • $\begingroup$ The same with solution space. Check what happens when the matrix is invertible. Then think what happens when the matrices have the "same" non trivial kernels $\endgroup$ – tst Aug 24 '18 at 10:35

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