# Characteristic and minimal polynomial in dual space

Let $$V$$ be a finite dimensional vector space over the field $$\mathbb{F}$$ and define a linear transformation $$T:V\rightarrow V$$.

We have that the dual space is $$V'=\text{Hom}(V,\mathbb{F})$$ and $$T':f \mapsto f \ \circ \ T$$, and know that if we have a basis $$\mathcal{B}$$ of $$V$$ then we can construct a dual basis $$\mathcal{B}'$$. I need to solve the question:

State a relationship between the characteristic polynomials of $$T$$ and $$T'$$, and the minimal polynomials of $$T$$ and $$T'$$. Explain your answer.

The matrix of $$T'$$ wrt $$\mathcal{B}'$$ is the transpose of the matrix of $$T$$ wrt $$\mathcal{B}$$. So my thinking is that because $$\det A=\det A^{tr}$$, we have that $$\chi_{T'}(x)=\chi_T(x)$$.

Is this correct? And how can I find a relationship between the minimal polynomials $$m_{T'}(x)$$ and $$m_T(x)$$?

Since, by definition $$\chi_T(\lambda)=\det(T-\lambda\operatorname{Id})$$, since a matrix and its transpose have the same determinants, and since$$(T-\lambda\operatorname{Id})^{\mathrm{tr}}=T^{\mathrm{tr}}-\lambda\operatorname{Id},$$$$T$$ and $$T'$$ have the same characteristic polynomials.
And if $$P(T)=0$$ for some polynomial $$P(x)\in\mathbb F[x]$$,then$$P(T^{\mathrm{tr}})=P(T)^{\mathrm{tr}}=0^{\mathrm{tr}}=0.$$So, $$T$$ and $$T'$$ have the same minimal polynomials.