Does the linear system in four variables determines a collection of three dimensional objects?

Question

We know that A linear system in three variables determines a collection of planes and the intersection point is the solution - ( https://en.wikipedia.org/wiki/System_of_linear_equations).

So, I want to know that what will the linear system in four variables determines? Is it a collection of three-dimensional objects and whose intersection will give the solution of the system?

Answer 1

In general a linear system may have zero, one or infinitely many solutions.

The case of zero solutions corresponds geometrically to the case of parallel lines in $\mathbb R^2$, planes in $\mathbb R^3$, etc.

The case of one solution corresponds always to a single point in $\mathbb R^n$.

The case with infinitely many solutions corresponds geometrically to the case of a line in $\mathbb R^2$, a plane or a line in $\mathbb R^3$, etc.

In general, for dimension higher than 3, the single equation of a linear system represents a hyperplane.

Answer 2

Every independent linear equation provides a restriction, so you lose 1 dimension. Therefore the dimension of solution space equals to the dimension of whole space minus number of independent equations.