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Monday
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August 26th
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Friday
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August 30th
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Fields, vector spaces (as having binary, unary, and nullary operations).
One thing slightly wrong, correcting it a homework question.
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Wednesday
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September 4th
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Subspaces, definition of independence/spanning/basis in terms of the
natural map from F^n being 1:1/onto/both.
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Friday
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September 6th
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If I is independent, and S (finite) spans, then we can find a basis
containing I, and inside I union S. Many many corollaries.
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Monday
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September 9th
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Exchange property for bases, therefore they all have the same size.
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Wednesday
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September 11th
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Nullity plus rank theorem, examples of 2x2 invertible matrices.
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Friday
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September 13th
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Systems of linear equations,
Lights Out
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Monday
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September 16th
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Hom(V,W), which Curtis calls L(V,W). Dual spaces.
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Wednesday
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September 18th
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Quotient spaces.
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Friday
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September 20th
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More quotient spaces, and a little bit on
cohomology of graphs (special topic).
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Monday
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September 23rd
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Direct sums of spaces and maps.
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Wednesday
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September 25th
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Double duals, dual bases, transposes.
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Friday
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September 27th
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Pairings between two vector spaces, and the relation to
dual spaces. Cokernels. Duality exchanges 1:1ness and ontoness.
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Monday and Wednesday
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September 30th, October 2nd
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Review for midterm #1.
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Friday
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October 4th
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Midterm #1.
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Monday
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October 7th
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Properties of determinant. Repeated rows gives 0 => switching rows negates.
Reverse implication also true, as long as 1+1 isn't 0.
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Wednesday
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October 9th
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Proof that there exists a unique function "det" with the given properties.
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Friday
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October 11th
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Eigenvalues. Algebraically closed fields.
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Monday
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October 14th
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Induced maps on quotient spaces by T-invariant subspaces.
Every matrix can be upper triangularized (over an algebraically closed field).
Generalized eigenspaces.
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Wednesday
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October 16th
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V is the direct sum of its generalized eigenspaces.
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Friday
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October 18th
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Nilpotent matrices. Statement of Jordan Canonical Form. Viewing a
nilpotent matrix as people walking off a cliff in a bunch of
independent queues.
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Monday
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October 21st
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More JCF: another construction of the good basis for nilpotent
transformations. Uniqueness of JCF.
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Wednesday
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October 23rd
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Yet more JCF. Every transformation is the sum of a diagonalizable
and a commuting nilpotent part.
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Friday
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October 25th
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Side topic: finitely generated modules over (certain) rings.
The classification of finite abelian groups.
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Monday
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October 28st
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Bilinear forms. Gram matrices. How Gram matrices change under change
of basis.
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Wednesday
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October 30th
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Hints on the homework. Diagonalizable matrices are those that
satisfy squarefree polynomials. "Nondegeneracy" of bilinear forms.
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Friday
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November 1st
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Bilinear forms and perps.
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M, W, F, W
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November 4th, 6th, 8th, 13th
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Reviewing for, taking, and going over the midterm.
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Friday
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November 15th
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Yet another proof of the decomposition into generalized eigenspaces.
Splitting spaces with a symmetric bilinear form into their radical
and a nondegenerate space.
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Monday
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November 18th
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Sylvester's Law of Inertia: bilinear forms over real or complex spaces
have orthogonal bases that are almost normal -- the norms can be
taken to be 0, 1, or -1.
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Wednesday
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November 20th
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The Clever™ algorithm for web searches.
See
the
original paper, in PostScript, or
in PDF.
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Friday
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November 22nd
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Introducing tensors.
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