Let $V$ and $W$ be finite dimensional vector spaces over $\mathbb{R}$ and let $T_{1} : V \rightarrow V$ and $T_{2} : W \rightarrow W$ be linear transformations whose minimal polynomials are given by $f_{1}(x)=x^{3}+x^{2}+x+1$ and $f_{2}(x)=x^{4}-x^{2}-2$. Let $T: V \bigoplus W \rightarrow V \bigoplus W$ be the linear transformation defined by $T(v,w)=(T_{1}(v),T_{2}(w))$ for $ (v,w) \in V \bigoplus W$ and let $f(x)$ be the minimal polynomial of $T$ . Then,
1) deg$f(x)=7$
2)deg$f(x)=5$
3) nullity$(T)=1$
4)nullity$(T)=0$.
what i know is minimal polynomial of $T$ will be $lcm \{ f_{1}(x),f_{2}(x)\}=lcm\{(x+1)(x^{2}+1),(x^{2}+1)(x^{2}-2)\}=(x+1)(x^{2}+1)(x^{2}-2)$ is it correct? and how to comment on nullity?
When you have the minimal polynomial with no factor of the form $x^k$ then cant you deduce the nullity of an operator?