# Conjugating matrices

This question is probably very elementary.

Basically I want to prove that conjugate matrices represent the same linear map but in different bases.

It is intuitively clear since if $M=X^{-1}NX$ and $N$ is expressed wrt basis $V$ then $X$ gives the coefficients of the linear combinations of vectors in $V$ for the new basis wrt which $M$ is expressed. So take a basis vector for $M$, then $X$ maps it to its representation wrt $V$ then $N$ acts on it then $X^{-1}$ maps it back to the expression wrt $M$'s basis.

But I don't know how to argue this rigorously. Maybe I also have to mention that $N$ is a linear map so this works? Please help!

Thanks.

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