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I know what happens in $\mathbb{R^2}$ and $\mathbb R^3$. In $\mathbb{R^2}$ , two lines either intersect in a point or they are parallel.

In $\mathbb{R^3}$, two lines (or a line and a plane) can intersect in a point, be parallel or be askew. Two planes either intersect (where the intersection is a line) or be parallel.

Now, can one infer what will happen in $\mathbb{R^4}$ merely by looking what happens in ${\mathbb{R^3}}$? What could happen between a $dim=2$ plane and a hyperplane? I think it's hard to find an intuitive answer because for example, in $\mathbb{R^4}$ two planes can have the same behaviour as in ${\mathbb{R^3}}$, but they can also intersect at one point. Is there a way to find this mathematically?

Giovanni
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markoff
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1 Answers1

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The question is a little vague, but I would say that yes, you can infer what happens in four dimensions.

The basic observation is that the intersection of two (affine) subspaces, if nonempty, is always an affine subspace. This can be more or less obvious depending on how you define affine subspaces.

In two dimensions, if you look at the relative position of a plane and a line, the only possibility is that the line lies on the plane, so the intersection is a line: this is because there is nothing outside the plane. You clearly can't have empty intersection, and if the intersection was somehow a single point (zero-dimensional), the line would have to get off the plane, so you'd need at least three dimensions.

In three dimensions, if you look at the relative position of two planes, then either the two are parallel or they are identical (so the intersection is a plane), or you can tilt one to get a line in the intersection. The two can't intersect in a single point: if they did, you would have four independent directions to move into from the point, which is not possible within a three-dimensional space.

In four dimensions, if you look at relative position of a $2$-plane and a $3$-subspace, the plane can either lie within the $3$-subspace, it can also lie within a parallel $3$-subspace, or you can tilt it, and then the intersection will be a line. It cannot possibly be a single point, as there are too few degrees of freedom: if you have a $2$-plane and a $3$-space intersecting in a single point, then you can move in $5$ different directions from the point.

More generally, if you look at $R^n$ and look at a $(n-1)$-hyperplane intersecting a $2$-plane, the intersection will be the whole plane, empty, or a single line, and yet more generally, if you look at the intersection of a $d_1$-dimensional and $d_2$-dimensional subspace, where $d_1\leq d_2<n$, it will either be empty or be a subspace of dimension between $d_1+d_2-n$ (or $0$ if this is negative) and $d_1$.

To deduce it mathematically, we simply assume that if the two subspaces intersect, they may as well intersect at the origin (this is always true, up to translation). Then we can look at the linear basis of the intersection, and then find vectors to supplement it to a basis of one space and then the other. By general linear algebra, all those vectors will be linearly independent (and form a basis of the linear span of the two spaces together), and therefore there can be only as many of those as the dimension of the enveloping space.

tomasz
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