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Majoring in Mathematics (view source)
Revision as of 02:46, October 16, 2009
, 02:46, October 16, 2009Linear algebra
# Bezout's theorem. Discuss intersection of conic sections, plane cubics, etc. Two conics intersect at 4 points, in generally.
# Bezout's theorem. Discuss intersection of conic sections, plane cubics, etc. Two conics intersect at 4 points, in generally.
# Is the quintic solvable in radicals? And what in the world does this problem have to do with group theory?
# Is the quintic solvable in radicals? And what in the world does this problem have to do with group theory?
# The ancient Greeks were fascinated by geometric constructions with straightedge and compass. They could bisect angles and take square roots, but they couldn't trisect angles or take cube roots. Why not?
A motivating question: failure of unique factorization in certain number fields (with explicit examples)
A motivating question: failure of unique factorization in certain number fields (with explicit examples)
=== Linear Algebra ===
If you throw a beach ball into the air, letting it spin in any way whatsoever, and then catch it, all of its rotations will comprise one rotation about some axis. That is, there will be two "poles" that will end exactly where they started, and all other points will just have rotated around them. What does this have to do with orthogonal operators? What does it have to do with the fact that every cubic equation has at least one real root?
If you do the same with a ring (e.g. a Hula Hoop), spinning it and letting it fall on its original "footprint", there will be no points that land exactly on their original position (except in the case in which it didn't move at all.) What does this have to do with the fact that quadratic equations are ''not'' guaranteed to have any real roots?
(Something about eigenvectors of distinct eigenvalues of an orthogonal/Hermitian matrix being orthogonal, can't think of a cute example just now.)
=== Geometry ===
=== Geometry ===