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Key ideas from last time

  1. A linear system of equations Ax = b can be reduced to row echelon form, whence the solutions can just be read off.
  2. Univariate polynomials have GCD's (Greatest Common Divisors) that can be computed by the Euclidean algorithm (though modular algorithms are significantly more efficient, and are what is actually implemented in most computer algebra systems).
  3. An ideal I is a subset of a ring which is closed under addition and by multiplication from elements of the ring. We can think of the GCD of a pair of univariate polynomials as a useful generator (or basis) for the ideal generated by the original polynomials.

Robert Corless
Tue Mar 12 21:09:19 EST 1996