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### III.17 Dimension

What is the difference between a two-dimensional set and a three-dimensional set? A rough answer that one might give is that a two-dimensional set lives inside a plane, while a three-dimensional set fills up a portion of space. Is this a good answer? For many sets it does seem to be: triangles, squares, and circles can be drawn in a plane, while tetrahedra, cubes, and spheres cannot. But how about the surface of a sphere? This we would normally think of as two dimensional, contrasting it with the solid sphere, which is three dimensional. But the surface of a sphere does not live inside a plane.

Does this mean that our rough definition was incorrect? Not exactly. From the perspective of linear algebra, the set {(x, y, z) : x ...

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