# To find minimal polynomial of a linear transformation

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 $$f_{1}(x)=x^{3}+x^{2}+1$$ 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$$(T)=0$$.