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Jun, 2019
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Linear Algebra

According to Dieudonné’s book [ Die89 ] , it was Emmy Noether who played a central role in formulating the vague idea of Poincaré on homology into a rigorous mathematics in 1930’s. According to the book on history of algebra [ vdW85 ] by van der Waerden, Noether also formulated linear algebra in the modern form.

We need linear algebra when we want to compute homology or cohomology with coefficients in a field, since linear algebra is nothing but theory of modules (of finite rank) over a field. We need dual vector spaces when we use the Kronecker product between homology and cohomology.

In order to construct important spaces, matrix groups are useful. Classical groups such as O ( n ) and U ( n ) and exceptional groups G 2 , F 4 , E 6 , E 7 , E 8 are good sources. We need, however, linear algebra over a “noncommutative field” or a “nonassociative field” when we study these groups.

A new and exciting variation of linear algebra is the study of “ field with one element 𝔽 1 ”. There seems to be a direct connection to the stable homotopy groups of spheres .

And we need linear algebra when we use vector bundles and K -theory .

We obtain the notion of matroids from linear dependencies of vectors. Matroids are used as a unifying language of various combinatorial structures such as hyperplane arrangements and graphs .