# Linear Space

1. Feb 4, 2009

### 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.

2. Feb 4, 2009

What do you mean, break down?

3. Feb 4, 2009

### 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?

4. Feb 4, 2009

Then you don't have a vector space.

5. Feb 4, 2009

### 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.

6. Feb 5, 2009

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.

7. Feb 5, 2009

### Office_Shredder

Staff Emeritus
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.

8. Feb 12, 2009

### 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).