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Let V = Pn(R), the vector space of all polynomials of degree at most n. And...

Let V = Pn(R), the vector space of all polynomials of degree at most n. And let T : V → V be a linear transformation. Prove that there exists a non-zero linear transformation S : V → V such that T ◦ S = 0 (that is, T(S(v)) = 0 for all v ∈ V) if and only if there exists a non-zero vector v ∈ V such that T(v) = 0.

Hint: For the backwards direction, consider building a basis for V that contains v (be sure to explain why you can always do this).

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