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Let $V$ and $W$ be finite-dimensional vector spaces over $R$ and let $T_1 \colon V \rightarrow V$ and $T_2 \colon W \rightarrow W$ 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 $T \colon V\oplus W \rightarrow V\oplus W$ be the linear transformation defined by $T(v,w)=(T(v),T(w))$ for $(v,w) \in V \oplus W$ and let $f(x)$ be the minimal polynomial of $T$. Then which of the following is true?

  1. $\deg f(x)=7$.
  2. $\deg f(x)=5$.
  3. $\operatorname{nullity}(T)=1$.
  4. $\operatorname{nullity}(T)=0$
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What is your question? –  Sasha Mar 20 '13 at 14:32
Choose the correct answers with suitabe reason. –  Kutubuddin Sardar Mar 20 '13 at 14:32
Note that $f_1(x) = (1+x)(1+x^2)$ and $f_2(x) = (1+x^2)(x^2-2)$ –  Sasha Mar 20 '13 at 14:37
You should first figure out why the minimal polynomial of $T$ divides the minimal polynomials of $T_1$ and $T_2$. –  Jim Mar 20 '13 at 15:21

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