what is degree of minimal polynomial?

Question

Let $V$ and $ W$ be finite dimensional vector space over $\mathbb R $ and let $T_1 : V \rightarrow V$ and $T_2 : W \rightarrow W$ be linear transformation whose minimal polynomial are $f_1 (x)= x^3+x^2+x+1$ and$f_2 (x)= x^4 - x^2-2$. let $T : V\oplus W \rightarrow V \oplus W$ be linear transformation s.t. $$T(v,w) =(T_1(v),T_2 (w)) $$ minimal polynomial of T is $f(x)$, then deg $f(x)$ =? and nulity T =?

I can't find such $T_1$, $T_2$and $T$ please guide me..

I don't know where to begin... I am stuck on this problem. Can anyone help me please?

Answer 1

Just to provide the obvious answer to this old question: the minimal polynomial of a linear operator that stabilises each of a pair of complementary subspaces is the (monic) least common multiple of the minimal polynomials of its restriction to those subspaces; this is immediate from the definition.

One easily computes $\gcd(x^4-x^2-2,x^3+x^2+x+1)=x^2+1$, and using the relation $\gcd(a,b)\operatorname{lcm}(a,b)=ab$, the least common multiple of these polynomials is therefore $(x^4-x^2-2)(x^3+x^2+x+1)/(x^2+1)=x^5+x^4-x^3-x^2-2x-2$.