Ellipse in vector space I cannot understand this (from https://en.wikipedia.org/wiki/Space_(mathematics) )

Linear operations, given in a linear space by definition, lead to such notions as straight lines (and planes, and other linear subspaces); parallel lines; ellipses (and ellipsoids). However, orthogonal (perpendicular) lines cannot be defined, and circles cannot be singled out among ellipses.

How can I define an ellipse in a vector space without using the notion of distance? And then, if I can define and ellipse, why I can't define a circle?
Thanks in advance. 
 A: Let $V$ be a two-dimensional real vector space. We can define an ellipse in $V$ to be a set of the form
$$ E(u,v) := \{ \cos \theta \cdot u + \sin \theta \cdot v \, | \, \theta \in [0,2\pi] \} $$
where $u,v \in V$ are two linearly independent vectors. This doesn't involve the notion of distance and if $T \colon V \rightarrow V$ is an invertible linear map, it will map an ellipse $E(u,v)$ to another ellipse $E(u',v')$ so this notion is invariant under linear maps.
If $V = \mathbb{R}^2$ then the set $E(u,v)$ really looks like an ellipse. If $u,v$ are perpendicular with respect to the standard inner product, then $E(u,v)$ will be an ellipse centered at the origin whose axes are $u,v$ (and if $\| u \| = \| v \|$ it will be a circle) but even if $u$ and $v$ are not orthogonal, only linearly independent, this set will look like an ellipse. But without introducing a notion of orthogonality and length, all the ellipses look the same and you "can't tell" which one is a circle and which one is a true ellipse.
This definition might look quite ad hoc so let me offer another definition that also works in all dimensions. If $V$ is a finite dimensional real vector space, define a set $E$ to be a (non-degenerate) ellipsoid if there exists a positive-definite quadratic form $q$ on $V$ and $c > 0$ such that $E = q^{-1}(0)$. This makes sense in any real vector space without any notion of distance or inner product. Taking this as the definition of an ellipsoid, one can then prove that if we endow $V$ with an inner product $\left< \cdot, \cdot \right>$, then any ellipsoid has $n$ orthogonal axes - this is the orthogonal diagonalization theorem for quadratic forms and using it one can recover the usual definition of an ellipsoid in $\mathbb{R}^n$. 
