Minimal polynomial and directsum of vector spaces $T_1,T_2$ be linear maps on $V$ and $W$ respectively which are finite dimensional vector space over Reals with minimal polynomials respectively $x^3+x^2+x+1,x^4-x^2-2$, let $T:V\bigoplus W\to V\bigoplus W, T(v_1,v_2)=(T_1(v_1),T_2(v_2))$ and let $f$ be the minpoly then 
$1. \text{ Deg}(f)=7$
$2. \text{ Deg}(f)=5$
$3. \text{ Nul}(T)=0$
$4. \text{ Nul}(T)=1$
Could anyone help me to solve this one? Guess: Minpoly for $T$ will be lcm of minpoly of $T_1,T_2$? 

For sure it will be the LCM because the matrix of $T$ in a suitable basis will be of the block form $\begin{pmatrix}A&0\\0&B\end{pmatrix}$
As a consequence, a polynomial $p$ annihilates $T$, iff it annihilates $T_1$ as well as $T_2$ iff the minimal polynomial of $T_1$ and that of $T_2$ both divide $p$.
So get the degree!
