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Exercise 4.5

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Proof. Define $Υ : P_{m} (𝔽) \to 𝔽^{m + 1}$ by

Υ (p) = (p (z_{1}), \dots, p (z_{m + 1}))

We have that $null Υ = {0}$ , because no polynomial in $P_{m} (𝔽)$ has $m + 1$ zeros. Hence $Υ$ is injective, proving the uniqueness part. By the Fundamental Theorem of Linear Maps, it follows that

\dim range Υ = \dim P_{m} (𝔽) = \dim 𝔽^{m + 1}

Therefore $Υ$ is surjective, proving the existence part. □

celiopassos

2017-10-06 00:00

Exercise 4.5

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