Intersections of axes and affine space

  • Thread starter jostpuur
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Let natural numbers [itex]N,M[/itex] be fixed such that [itex]1\leq M < N[/itex]. If [itex]x\in\mathbb{R}^N[/itex] is some vector, and [itex]V\subset\mathbb{R}^N[/itex] be some subspace with [itex]\textrm{dim}(V)=M[/itex]. How likely is it, that [itex]x+V[/itex] intersects the axes [itex]\langle e_1\rangle,\ldots, \langle e_N\rangle[/itex] somewhere outside the origin?

I mean that [itex]x+V[/itex] intersects the axis [itex]\langle e_n\rangle[/itex] iff there exists [itex]\alpha\in\mathbb{R}[/itex] such that [itex]\alpha e_n\in x + V[/itex] and [itex]\alpha \neq 0[/itex].

By "likely" I mean that for example if [itex]x[/itex] is a sample point of some random vector, and if [itex]V[/itex] is spanned by some [itex]M[/itex] sample vectors, and if the random vectors in question can be described by non-zero probability densities, then what is the probability for intersections of [itex]x+V[/itex] and [itex]\langle e_n\rangle[/itex] to exist?

Example M=1, N=2. If we draw a random line on the plane, chances are that the line will intersect both axes. The outcome that line intersects only one axis, is a special case, which can occur with zero probability.

Example M=2, N=3. If we draw a random plane into three dimensional space, chances are that the plane will intersect all three axes. The plane can also intersect only two or one axes, but these are special cases, which can occur with zero probability.

Example M=1, N=3. If we draw a random line into three dimensional space, changes are that the line will miss all three axes. The line can intersect one or two axes, but these outcomes occur with zero probability. The outcome that the line would intersect all three axes is impossible.

Looks complicated! I don't see a pattern here. What happens when [itex]N>3[/itex]?
 

HallsofIvy

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Before you can ask any question about a "random vector" you will have to specify what that means- in particular what probability distribution the "random" vector will satisfy.
 
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I was unable to prove this but I figured out sufficient rough reasoning, that convinced me, that the answer is this: If M=N-1, the affine subspace will intersect all axes with probability 1, and if M<N-1, the affine subspace will miss all axes with probability 1.

HallsofIvy's comment is not on right track. The probability densities don't need to be defined with a greater precision than what I already did. Look at the three low dimensional examples for example. It should be clear that the stuff works out like I said.
 

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