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$T_1,T_2$ 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

$1. Deg(f)=7$

$2. Deg(f)=5$

$3. Nul(T)=0$

$4. Nul(T)=1$

Could anyone help me to solve this one? Guess: Minpoly for $T$ will be lcm of minpoly of $T_1,T_2$?

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1 Answer 1

up vote 1 down vote accepted

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!

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Aneesh degree will be $5$ could you tell me why option $3$ is true? –  Une Femme Douce Jun 4 '13 at 18:51
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0 is not an eigenvalue! (easy to check: the minimal polynomial has as roots the eigenvalues - counted without multiplicities) So $T$ is injective and does not have nullity!! So 3 is true –  Host-website-on-iPage Jun 7 '13 at 6:35

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