How can we prove that the dimension of a plane is 2?

Question

By the question, I mean that what is the theoretical proof of it? How can we prove that the vector space of displacements take place in a plane is a 2-dimensional vector space?

Answer 1

The elements of "the plane" are points. Assuming that you are talking about a Cartesian plane, you can describe your points by two coordinates $(x,y)$. We can write any point by a linear combination of the points $(1,0)$, $(0,1)$ (i.e., $(x,y) = x(1,0) + y(0,1)$). Hence, $\{ (1,0), (0,1) \}$ is a generator set of the plane.

Any set with less elements (in that case, an unitary set) can not generate the plane. For example, if $\{ (x,y) \}$ is a potential generator set for the plane, we can't write the point $(x+1,y)$ as a linear combination of $(x,y)$.

So a set with less than two elements is not a generator set of the plane and a set of more than two elements is not linear independent (any third point in the set $\{ (1,0), (0,1) \}$ can be written as a linear combination of that two points), and that suffices to prove that the dimension of a plane is $2$.

(Sorry for the bad english, I'm still learning the language)