# K-Theory also is a generalization of Linear Algebra.

K-Theory studies *families* of vector spaces, called *vector bundles*.

Examples:

There is only one kind of family over a single point; i.e., a single vector space.

There is also only one kind of family over the reals; i.e.,
the product family, V x **R**
→ **R**.

But there are two kinds of families over the circle, S^{1}:

## For more, see:

M. Walker with Eric Friedlander, * Rational Isomorphisms between K-theories and cohomology theories*,
Invent. Math., 154 (2003) 1-61.

## Note:

## Vector bundles give rise to rings and modules!

- Let E → X be a vector bundle over a space X
- Let R(X, k) be the set of maps to a field k
- Let H
^{0}(E/X) be the set of sections; i.e.,
the set of maps X → E such that
the composition X → E
→ X is the identity
- Then R(X, k) is a commutative ring and H
^{0}(E/X) is an
R(X, k)-module!

## But what about Algebraic Geometry?