This is because you weren't algebraing, you were pre-Hilberting :)
Let's forget we're in finite dimension for a minute, in which case it is not trivial to expect a scalar product on our vector space.
So now we have two things, which are separate: a vector space, and a scalar product.
The vector space says: you can sum my elements, multiply them by a scalar, and I'm stable by linear combinations. That's all it says, really.
Everything else, the geometry, is a consequence of the scalar product. It even defines the norm in the first place, thus the topology. For instance, there's no talking about convergence in a vector space if you don't have a norm. You can't even tell if a sequence converges.
You can define a norm without a scalar product, of course. But then you lose certain notions, like orthogonality.
Anyways, in the case of finite dimension vector spaces, there's always a scalar product at hand, and this is where the geometry comes from, really. The algebra portion of it - being a group with a scalar multiplication - is certainly necessary, but not the one to define geometry.
Let's forget we're in finite dimension for a minute, in which case it is not trivial to expect a scalar product on our vector space.
So now we have two things, which are separate: a vector space, and a scalar product.
The vector space says: you can sum my elements, multiply them by a scalar, and I'm stable by linear combinations. That's all it says, really.
Everything else, the geometry, is a consequence of the scalar product. It even defines the norm in the first place, thus the topology. For instance, there's no talking about convergence in a vector space if you don't have a norm. You can't even tell if a sequence converges.
You can define a norm without a scalar product, of course. But then you lose certain notions, like orthogonality.
Anyways, in the case of finite dimension vector spaces, there's always a scalar product at hand, and this is where the geometry comes from, really. The algebra portion of it - being a group with a scalar multiplication - is certainly necessary, but not the one to define geometry.