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I am self-studying Euclidean geometry, and I am a little confuse about the following statement.

In dimension bigger than three is possible two planes have exactly one point in common.

It is from a book written in Portuguese. How is that possible?

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    We can actually make an even stronger statement (Ross's answer demonstrates quite nicely how to prove it). In an $n$-dimensional space, given *any* integer $0\le m\le\frac{n}2$, there exist two $m$-dimensional subspaces whose intersection is precisely the origin.2012-10-17
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    Indeed, if $r+s\le n$, there's an $r$-dimensional space and an $s$-dimensional space whose intersection is just the origin.2012-10-17

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Think of $\mathbb R^4$. For one example, one plane is all the points $(x,y,0,0)$. The second is all points $(0,0,z,w)$. The intersection is the origin.

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    Dear Ross, but I do not have coordinates so far. I am allowed to use only the axioms. At least I understood that way.2012-10-17
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    @spohreis: I'm not sure what axioms you have that define the dimension, nor how they define a plane. I thought you were just looking for an example. Can you take four vectors that are a basis, define the plane from each of two pairs, and use linear independence to show the origin is the only common point?2012-10-17
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    Ok! It was a comment in the book. I do not have an axiom to define the dimension of the space so far.2012-10-17