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Varieties II: Quasiprojective Varieties

August 22, 2009

This post contains the definition of (quasi-) projective varieties, regular functions, morphisms, projective coordinate ring, the Nullstellensatz. It also contains the global section of sheaf of regular function over distinguished open sets and the fact that affine neighborhoods form a basis for the Zariski topology

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Varieties I: affine varieties

August 17, 2009

This post contains the defintiion of affine varieties, regular functions, coordinate ring, the Nullstellensatz, and the quotient of an affine variety upon action of a finite group. Read more…

Similarity and normal forms

August 2, 2009

This post will prove the uniqueness of Jordan form, Smith normal form and Rational canonical form in an equivalence class of similar matrices. A criterion to check similarlity of matrices using these forms is given, and a nice lemma of similarity being irrelevant to field extension is shown using rational canonical form.

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Polar decomposition

July 30, 2009

This post proves the existence of polar decomposition. Read more…

Rational Canonical Form

July 28, 2009

This post proves the existence of rational canonical form, using the structure theoerm of finite modules over PID. Read more…

Smith Normal Form

July 28, 2009

This post proves the existence of Smith normal form. The proof is pretty much the same as the one of reduced row echelon form.

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Jordan form II: Computations

July 26, 2009

This post talks about computation of Jordan form and Jordan basis, along with a few examples.

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