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)?

$f(x)=l.c.m(f_1(x),f_2(x))=(1+x^2)(1+x)(x^2-2)$ which is minimal polynomial of T.From the minimal polynomial we can say that 0 is not eigenvalue of T and hence T is invertible.Hence nullity(T) is 0.
