# Higher-dimensional spaces curiosity

## 1 Answer

\[a_1x+b_1y+c_1z+d_1w=e_1 \text{and} a_2x+b_2y+c_2z+d_2w=e_2.\]

It is possible for these two hyperplanes to have (I) no intersection or (II) coincide. However, if cases (I) and (II) does not happen, then using basic facts from linear algebra (you may wire $w$ in terms of the other three variables in one equation and replace in the other) one can conclude that the intersection will be a two-dimensional plane.

You conjecture is true in any dimension $n$, and can be proved with a very similar argument.

Poincare

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