Higher-dimensional spaces curiosity 

If the intersection of lines in R² is a point and that of planes in R³ is a line, is the intersection of 3D spaces in R⁴ a plane, that of 4D spaces in R⁵ a three-dimensional space and so on ?

1 Answer

I guess by intersection of 3D space in $R^4$ you mean intersection of two ahyperplanes: 
\[a_1x+b_1y+c_1z+d_1w=e_1   \text{and}  a_2x+b_2y+c_2z+d_2w=e_2.\]
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. 

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