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TIFR Mathematics 2022 | Part A | Question-5

Let $n$ be a positive integer, and let $V=\{f \in \mathbb{R}[x] \mid \operatorname{deg} f \leq n\}$ be the real vector space of real polynomials of degree at most $n$. Let $\operatorname{End}_{\mathbb{R}}(V)$ denote the real vector space of linear transformations from $V$ to itself. For $m \in \mathbb{Z}$, let $T_{m} \in$ End $_{\mathbb{R}}(V)$ be such that $\left(T_{m} f\right)(x)=f(x+m)$ for all $f \in V$. Then the dimension of the vector subspace of $\operatorname{End}_{\mathbb{R}}(V)$ given by $$\operatorname{Span}\left(\left\{T_{m} \mid m \in \mathbb{Z}\right\}\right)$$ is

  1. $1$
  2. $n$
  3. $n+1$
  4. $n^{2}$
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