What is the difference between algebra and linear algebra?

Linear Algebra is a special "subset" of algebra in which they only take care of the very basic linear transformations. There are many many transformations in Algebra, linear algebra only concentrate on the linear ones.

We say a transformation T: A --> B is linear over field F if

T(a + b) = T(a) + T(b) and kT(a) = T(ka)

where a, b is in A, k is in F, T(a) and T(b) is in B. A, B are two vector spaces.