- #1
Gjmdp
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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.
Thank you.
Thank you for your answer. How can you prove that?fresh_42 said: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##.
Your reasoning is correct if those assumptions are given.Gjmdp said:Thank you for your answer. How can you prove that?
I think more usual terms for regular are invertible or nonsingular.fresh_42 said:If ##n=m## and ##M## is regular and ##Mu=Mv## then ##u=v##.
##Mu = Mv \Rightarrow Mu - Mv = 0 \Rightarrow M(u - v) = 0##Gjmdp said:Thank you for your answer. How can you prove that?
The Paradox of u=v is a mathematical concept that states that if u=v, then v=u. This may seem like a simple and obvious statement, but it can lead to some confusing and contradictory conclusions when applied to certain equations and situations.
The Paradox of u=v is relevant in science because it highlights the importance of understanding and carefully considering the assumptions and logic behind mathematical equations and theories. It also challenges scientists to think critically and creatively when faced with seemingly contradictory information.
One example of the Paradox of u=v in science is the equation for calculating the speed of light, which states that the speed of light in a vacuum is equal to the distance traveled divided by the time it takes to travel that distance. This equation can be rearranged to show that the distance traveled is equal to the speed of light multiplied by the time it takes to travel that distance, highlighting the paradoxical nature of the equation.
The Paradox of u=v can be solved by carefully examining the assumptions and logic behind the equations and theories involved. This may involve redefining terms, considering alternative explanations, or developing new mathematical frameworks to better explain the phenomenon.
The Paradox of u=v highlights the importance of critical thinking and careful analysis in scientific research. It also encourages scientists to question and challenge established theories and assumptions, leading to new and innovative discoveries and advancements in the field.