Exploring Linear Space in the Real & Complex Planes

[Numeralysis]

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 not too sure how they would.

 

[adriank]

What do you mean, break down?

 

[Numeralysis]

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?

 

[adriank]

Then you don't have a vector space.

 

[Numeralysis]

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.

 

[adriank]

They are included because they are useful; there are many interesting theorems about vector spaces, and there are lots of things that can be modeled by vector spaces. If you omit some axioms such as 1v = v or 0 + v = v, then many of the theorems fail to be true.

 

[Office_Shredder]

They seem pretty intuitive, because every object that has been introduced to you as a linear space has those properties. If you have a set with an operation, we almost always use 0 and 1 to be defined as the additive and multiplicative identities; so if you had a set with an operation that had no such identity, we wouldn't call elements 0 and 1.

Off the top of my head I'm not able to think of a space that satisfies every vector space axiom except for those two.

 

[LogicalTime]

The statements 0+v= v and 1v=v are really saying that there are additive and multiplicative identities and telling you particularly what they are (0 & 1).