minimal polynomial of linear transformation Let $V$ and $W$ are finite dimensional vector spaces over $\mathbb{R}$ and let $ T_1:V\to  V$ and $ T_2:W\to W$ be linear transformations whose minimal polynomials are given by
$ f_1(x)=x^3+x^2+x+1 , f_2(x)=x^4-x^2-2$.
Let $ T:V\oplus W\to V\oplus W$ be defined by 
 $ T(v,w)=( T_1(v),T_2(w))$ $  for  (v,w)\in V\oplus W$ and let f(x) be minimal polynomial of T then how to find nullity of T and degree of f(x)?

Let The nullity of $T_i$ be $N_i(T_i)$ for $i=1,2$. Then the nullity of $T$ is: $N_1(T_1)\oplus N_2(T_2)$.  The minimal polynomial of $T$ is $lcm[f_1,f_2]$. Both of $T_1$ and $T_2$ are invertible because: $$T_1^3+T_1^2+T_1+I=0$$ From here: $$T_1(-T_1^2-T_1-I)=(-T_1^2-T_1-I)T_1=I_v$$  Similarly: $$T_2(\frac12(T_2^3-T_2))=(\frac12(T_2^3-T_2))T_2=I_w$$ So, the nullity of $T_1$ are $\{0_v\}$ and the nullity of $T_2$ is $\{0_w\}$. Thus, the nullity of $T$ is: $(0_v,0_w)$.
