Paradox about u=v

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  • Thread starter Gjmdp
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  • #1
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Main Question or Discussion Point

Let u, v be column vectors n x 1 and M a m x n matrix over a field K. If M*u= M*v, then (M^-1)*M*u=(M^-1)*M*v, thus, I*u=I*v. Hence u=v. But that shouldn't be the case. What is wrong in my reasoning?
Thank you.
 

Answers and Replies

  • #2
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There is no inverse matrix for ##M## in case ##n\neq m##. If ##n=m## and ##M## is regular and ##Mu=Mv## then ##u=v##.
 
  • #3
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There is no inverse matrix for ##M## in case ##n\neq m##. If ##n=m## and ##M## is regular and ##Mu=Mv## then ##u=v##.
Thank you for your answer. How can you prove that?
 
  • #4
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Thank you for your answer. How can you prove that?
Your reasoning is correct if those assumptions are given.
 
  • #5
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If ##n=m## and ##M## is regular and ##Mu=Mv## then ##u=v##.
I think more usual terms for regular are invertible or nonsingular.
Thank you for your answer. How can you prove that?
##Mu = Mv \Rightarrow Mu - Mv = 0 \Rightarrow M(u - v) = 0##
Assuming M is invertible, then ##M^{-1}M(u - v) = M^{-1}0 = 0##, or ##u - v = 0##
 

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