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, S¹:

There is the trivial one (i.e., the product family, V x S¹ → S¹).
But what is the other?

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 H⁰(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 H⁰(E/X) is an R(X, k)-module!

Question:

But what about Algebraic Geometry?