Prove Vector Equations: AB=Im & m<=n

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SUMMARY

The discussion centers on proving vector equations involving matrices A and B, where A is an mxn matrix and B is an nxm matrix, with the condition that AB = Im. The user successfully demonstrates that there is only one solution to the equation Bx = 0, confirming that x must equal 0, thus establishing the trivial solution. Furthermore, the user proves that m must be less than or equal to n, as having m greater than n would lead to infinite solutions for Bx = 0, contradicting the initial condition.

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Homework Statement


A is a mxn matrix and B is a nxm matrix. AB = Im.
a) Prove that there's only one solution to Bx = 0 where x and 0 are coloum vectors.
b) Prove that m<=n


Homework Equations


if A is a mxn n>m then there're infinite solutions to Ax=0


The Attempt at a Solution


a) Bx=0 => ABx=A0 => Ix=0 => x=0 so there's only the trivial solution.
b) according to the equation above if m>n then there would be an infinite amount of solutions to Bx=0 and not only one.
Did I do that right? This was a problem on an exam and it relative to the other questions it looks to easy.
 
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Hm, looks okay.
 

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