I am studying the basics of geometrical control theory, and I am struggling with some concepts. At the moment I am studyng the concept of distribution.
So far I have understood that a distribution is a law that associates to each point $x$ a subspace of the teangent space of $x$ :
$$\Delta : x \rightarrow \Delta (x)\subset T_x\mathbb{R}^{n}$$
but I cannot grasp the concept. For example, if I consider a distribution:
$$\Delta (x)=\begin{pmatrix} x_1 & 1\\ x_1x_3 &x_1 \\ 0 &0 \end{pmatrix}$$
if I consider the definition, it should associate to each point a subspace, but what does it mean?
Maybe each column of the distribution if a vector, and so a collection of vectors defines a subspace? This that I just said is just a reasoning I did, so I am not sure.
Moreover, I have studied that a distribution is given by a set of independet vectors:
$$\Delta (x)=\operatorname{span}[f_1(x),....,f_n(x)]$$
which I think is has to be true, otherwise they won't define a space. But I am also confused by the fact that each vectors is associated to a point, so if I take each vector by itself, I have a space with more vectors associated to points.
After this, the notes of my professor start describing costant rank distributions and integrable distributions, which are hard to understand for me at this point, considering that I have not clear the concept of distribution.
Can somebody please help me?