How is a plane mathematically described in a finite-dimensional Euclidean space?

Question

Mathematics

Question for mathematicians (geometers, probably) about how to set an M-dimensional plane (M <= N) in an N-dimensional Euclidean space.
I understand this at the level of 3-dimensional space: that
(1) there is a "basic" plane, given through 3 "basic" variables (x, y, z),
(2) there is a line at the intersection of 2 planes, given as a system equations of 2 planes,
(3) is a point lying at the intersection of 3 planes, given as a system of equations of 3 planes or as a vector (x, y, z).
Is it true that the plane of dimension Nk is given by a system of k+1 equations (0 <= k < N-1)? Is this true for Euclidean spaces of higher dimension?
Where can I read about this, briefly and with a presentation on the example (or by analogy) of 3-dimensional space?
ps Is a point considered a plane of zero dimension?

Answer 1

Here .