Why is Null Space the basis for a plane in 3D space?

Question

Geometry

Preface: By Null space I mean the set of solution vectors of the x-owls in the equation Ax = b (where A is a matrix and x and b are vectors), when given solutions (x's), the vector b is calculated into a null vector. This seems to be called the core of the linear space .
Here is the equation of the plane x-2y+3z=0, in the "answer" the basis is the following two vectors:

|1 -2 3| **************************|2|**********|-3|
|0 0 0| - Null space— Xn = c1 |1| + c2 | 0|
|0 0 0|******************************|0|******* *|1|

The coefficients can be set to unity and we get the basis. Why is that? Why are the solutions in which the given matrix is ​​calculated into a zero vector the basis of the plane?

Answer 1

The plane passes through zero. This means that the basis vector in this plane also satisfies the equation (after all, any vector in the plane has the coordinates of the end point - and it is on the plane).
Your matrix with zeros when multiplied just calculates the equation. And since the vector satisfies the equation, it will be 0.