Proof.

Theorem AIU (Additive Inverses are Unique) Suppose that V is a vector space. For each u∈V, the additive inverse, −u, is unique.

Proof.

As obvious as the next three theorems appear, nowhere have we guaranteed that the zero scalar, scalar multiplication and the zero vector all interact this way. Until we have proved it, anyway.

Theorem ZSSM (Zero Scalar in Scalar Multiplication) Suppose that V is a vector space and u∈V. Then 0u=0.

Proof.

Here’s another theorem that looks like it should be obvious, but is still in need of a proof.

via Vector Spaces.