minimal polynomial is independent of choice of basis Say I have a linear map $T:V\to V$, and two matrix representations, $M$, $N$, say, such that there exists an invertible matrix, $P$, with the same dimension and $M=P^{-1}NP$. Say $m_M$ and $m_N$ are minimal polynomials for $M$ and $N$. How would I show that $m_N(M)=0$? 
Am I allowed to say that $m_N(M)=m_N(P^{-1}NP)= m_N(P^{-1})m_N(N)m_N(P)=0$ and that $m_M(N)=0$ follows similarly and hence the minimal polynomial is independent of choice of basis?
 A: $M=P^{-1}NP$ implies $M^k=P^{-1}N^kP$ for all $k$ and so $f(M)=P^{-1}f(N)P$ for all polynomials $f$. Therefore, the set of polynomials that kill $M$ coincides with the set of polynomials that kill $N$.
A: The way the question is phrased, I assume the intention was to bring $T$ into the argument in some way. And assuming that $M,N$ are matrices of $T$ with respect to two different bases (I think that is what you mean by "representation" here), that is indeed the simplest approach.
The crucial point is that one can define the minimal polynomial of $T$ itself, without ever using matrices: it is the minimal degree monic polynomial $X^d+c_{d-1}X^{d-1}+\cdots+c_1X+c_0$ such that one has
$$T^d+c_{d-1}T^{d-1}+\cdots+c_1T+c_0I=0\in\mathcal L(V,V).\tag1$$
(That some such monic polynomial exists is a consequence of the finite dimensionality of $\mathcal L(V,V)$, and that there is a unique one of minimal degree, which moreover divides all others, is the standard fact that every nonzero ideal of $K[X]$ has a unique monic generator; these arguments are identical for what one uses to define minimal polynomials of square matrices.)
What remains is to show that if $A$ is a matrix of $T$ with respect to some basis of$~V$ then $(1)$ is satisfied if and only if
$$A^d+c_{d-1}A^{d-1}+\cdots+c_1A+c_0I=0\in\operatorname{Mat}_{n,n}(K),\tag2$$
which is immediate from the way matrix arithmetic corresponds to arithmetic on linear operators. Then the minimal polynomial of $A$ is the same as that of $T$, and in the question $M,N$ are examples of such matrices$~A$.
