Complete the proof to the right that the zero vector is unique Axioms In the fol

Question

Complete the proof to the right that the zero vector is unique Axioms In the following axioms, u, v, and w are in vector space V and c and d are scalars. Suppose that w in V has the property that u +w-w 0+w w+0by Axiom 1, and w + 0-wby Axiom Type whole numbers.) ll u in V. In particular, 0+ w o. But . Hence, w-w+0-o-w-o 1. The sum uv is in V 3. (u+v)+w ut(v+ w) 4. Vhas a vector 0 such that u + 0 = u 5. For each u in V, there is a vector -u in V such that u-u)-0. 6. The scalar multiple cu is in V. 8. (c+d)u cu+ du 9, c(du) = (cd)u 0 u-u

Explanation / Answer

For u,v lying in vector space V, by axiom 2, u+v = v+u.

Thus, 0+w = w+0 is by axiom [2].

Again by axiom 4, V has a vector 0 (a zero vector), so that,. u+0 = u

Thus, w+0 = w is by axiom [4].

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