| Suppose Q and R are lines in the plane. Typically, Q and R do intersect in a point. |
| How do their codimensions add? |
| (2 - dim(Q)) + (2 - dim(R)) | = (2 - 1) + (2 - 1) = 2 |
| = 2 - 0 = 2 - dim(Q ∩ R) |
|
| That dim(Q ∩ R) = 0 reinforces the observation that Q and R intersect in
a point, which has dimension 0. |
| |
| Suppose P is a point in the plane and Q is a line in the plane.
Typically, P and Q do not intersect. |
| How do their codimensions add? |
| (2 - dim(P)) + (2 - dim(Q)) | = (2 - 0) + (2 - 1) = 3 |
| = 2 - (-1) = 2 - dim(P ∩ Q) |
|
| That dim(P ∩ Q) = -1 expresses the condition that typically P and Q
do not intersect. |
| |
| From these examples we hypothesize that if A and B lie in n-dimensional space, |
| n - dim(A ∩ B) = (n - dim(A)) + (n - dim(B)) |
| That is, |
| codim(A ∩ B) = codim(A) + codim(B) |
| |
| Exercise: verify this rule for the intersections of lines, planes, and points in
3-dimensional space. |