Below is a seven-step proof of part (b) of Theorem 4.1.1. Justify each step either by stating that it is true by hypothesis or by specifying which of the ten vector space axioms applies Hypothesis: Let u be any vector in a vector space V, let 0 be the zero vector in and let k be a scalar.
Conclusion: Then k0 = 0.

Proof: (1) k₀+ku = k(0+u)
(2) = ku
(3) Since ku is in V. -ku is in V
(4) Therefore, (k0+ku)+(-ku) = ku+(-ku)
(5) k0+(ku+(-ku) = ku + (-ku)
(6) K0+0 = 0
(7) k0 = 0