

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
.


