# A Counter Examples in Linear Algebra (Vector Space)

I have been studying linear algebra for a few years now and in fact, also teaching it. What makes learning (and teaching) mathematics more interesting is to find examples and/or counterexamples of what we learn. As a process, I am trying to find counterexamples of sets along with two operations $$+$$ and $$\cdot$$, which we will call for the time being "addition" and "scalar multiplication" which do not form a vector space because they fail to satisfy exactly one of the axioms of a vector space.

If I am to list out the axioms and hence the definition, it proceeds as follows:-

A set $$V$$ along with two operations $$+: V \times V \rightarrow V$$ and $$\cdot: \mathbb{F} \times V \rightarrow V$$, where $$\mathbb{F}$$ is a field, is called a "vector space" over the field $$\mathbb{F}$$ if:

1. $$\forall x, y, z \in V$$ we have $$\left( x + y \right) + z = x + \left( y + z \right)$$
2. $$\forall x, y, \in V$$ we have $$x + y = y + x$$
3. $$\exists 0 \in V$$ such that $$\forall x \in V$$, we have $$x + 0 = x$$
4. $$\forall x \in V$$, $$\exists y \in V$$ such that $$x + y = 0$$
5. $$\forall x, y \in V$$ and $$\forall \alpha \in \mathbb{F}$$, we have $$\alpha \cdot \left( x + y \right) = \alpha \cdot x + \alpha \cdot y$$
6. $$\forall x \in V$$ and $$\forall \alpha, \beta \in \mathbb{F}$$, we have $$\left( \alpha + \beta \right) \cdot x = \alpha \cdot x + \beta \cdot y$$
7. $$\forall x \in V$$ and $$\forall \alpha, \beta \in \mathbb{F}$$, we have $$\alpha \cdot \left( \beta \cdot x \right) = \left( \alpha \beta \right) \cdot x$$, where $$\alpha \beta$$ denotes the multiplication of $$\alpha$$ with $$\beta$$ in the field $$\mathbb{F}$$
8. $$\forall x \in V$$, we have $$1 \cdot v = v$$, where $$1 \in \mathbb{F}$$ is the unity.

It is not so difficult (if not easy) to find counterexamples of sets, fields and operations which satisfy all but one property from $$1$$ through $$7$$. However, I have not yet been able to find an example which satisfy all properties except $$8$$ and hence fails to be a vector space. I would like some help in constructing such a counter example.

Take any nontrivial abelian group $$V$$ and any field $$\mathbb{F}$$, and define $$c\cdot v = 0_V$$ for all $$c \in \mathbb{F}$$ and $$v\in V$$.