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TIFR2021-Maths-A: 12

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Let $V$ be a vector space over a field $F$. Consider the following assertions:

  1. $V$ is finite dimensional
  2. For every linear transformation $T:V\rightarrow V$, there exists a nonzero polynomial $p(x)\in F[x]$ such that $p(T):V\rightarrow V$ is the zero map.

Which one of the following statements is correct?

  1. $(\text{I})$ implies $(\text{II})$ but $(\text{II})$ does not imply $(\text{I})$
  2. $(\text{II})$ implies $(\text{I})$ but $(\text{I})$ does not imply $(\text{II})$
  3. $(\text{I})$ implies $(\text{II})$ and $(\text{II})$ implies $(\text{I})$
  4. $(\text{I})$ does not imply $(\text{II})$, and $(\text{II})$ does not imply $(\text{I})$
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