Question

Let V be a finite-dimensional vector space and let T $\in$ L(V) be an operator. In this problem you show that there is a nonzero polynomial $p \in P(F)$ such that p(T) = 0.

(a) What is 0 in this context? A polynomial? A linear map? An element of V?

(b) Define by . Prove that $\phi$ is a linear map.

(c) Prove that if where V is infinite-dimensional and W is finite-dimensional, then S cannot be injective.

(d) Use the preceding parts to prove there is a nonzero polynomial $p \in P(F)$ such that p(T) = 0.

o : P(C) → L(V) o(p) = p(T) S E (V, W)

Answer 1

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