Problem in linear algebra. Intersection of planes at a point?

Question

Linear algebra

For some reason I really liked this puzzle. I tried several ways to solve it and had a lot of fun in the process.

What is the smallest number of space dimensions in which two planes can intersect at a point? (We are talking about linear space)

Taken from Gelfand I.M. — Lectures on linear algebra.

Answer 1

In order for the planes to intersect at a point, one of them must be rotated slightly along the time axis so that it turns into a straight line that sends the two halves of this very plane into the past and the future. Then the point of intersection of this line will be the desired point. Something like this?

Answer 2

I'll try to write an analytical answer.
N=1 - no planes
N=2 - all planes coincide, that is, they have an infinite number of intersection points.
N=3 - the planes are either parallel or intersect in a straight line.
Let's prove it.
Take two intersecting planes in three-dimensional space. Their equations can be written like this:

A1*x+B1*y+C1*z = D1.

A2*x+B2*y+C2*z = D2.

When rotated, the number of intersection points will not change, so we can rotate the planes so that A1 is non-zero. Then:

x = (D1-B1*y-C1*z )/A1,
A2*(D1-B1*y-C1*z )/A1+B2*y+C2*z = D2.

Let the planes intersect at the point (x0,y0,z0), that is, this point is the solution of the two higher equations. Let's substitute y0+1 into the equations, and solve the system (I won't continue the calculations, I think it's pretty obvious). We get the second point of intersection. The only assumption we made was that our two planes intersect at least at one point. There are always more than one such points, if any, that is, N=3 is not a solution to the problem.
N=4
In four-dimensional space, a plane is defined by a system of two linear equations in four variables: x, y, z and t.
Let us give an example when two planes intersect at one and exactly one point. The first plane is formed by the abscissa and ordinate axes:

z=0,
t=0. 

The second plane is formed by the applicate axis and the axis traditionally called the time axis:

x=0,
y=0.

In four-dimensional space, only one point is on both of these planes (satisfies both systems of equations). This is the origin (0,0,0,0), so the answer is 4 .
Obviously, for N>4, one can take a four-dimensional hyperplane, and the above example will work in it.