Exactly, but, I'm wondering when does this definition not hold true. These two properties seem pretty obvious, and pretty intuitive. More than anything, I'm wondering why are they included when defining a Linear Space. Other than for extra-proofing.
And if these definitions fail.
The two pretty obvious & intuitive properties. Where they don't work anymore; such that when you have v living in V and you multiply 1 by v, it longer equals v or add the zero vector it doesn't equal itself?
I was wondering, some of the things that define a Linear Space such as:
v \in V then 1v = v or \vec{0} \in V such that \vec{0} + v = v
They seem very obvious and intuitive, but, is there ever a time they break down in the Real plane? I think they might break down in the complex plane, but, I'm...