Prove that the operation of linear combination, as in Definition 2.2.7, makes into an -dimensional vector space over . The zero vector is the infinitesimal curve represented by the constant . If , then where , defined for all sufficiently small values of .

Proof:

We verify the axioms of a vector space.

Multiplicative axioms:

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Additive Axioms:

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Hence .

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Distributive Axioms:

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Hence is a vector space over . Since , is -dimensional.