Your language?
Feb, 2020
Sun Mon Tue Wed Thu Fri Sat
2 3 4 5 6 7 8
9 10 11 12 13 14 15
16 17 18 19 20 21 22
23 24 25 26 27 28 29

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 .