What is the difference between Linear Algebra and Vector Algebra?

I'm not sure that "Vector Algebra" is really a subject or branch of mathematics, but if it is it probably refers to the arithmetic of vectors in two or three dimensions, as is used in elementary physics.  But "Linear Algebra" is a vast generalization of that.  Linear Algebra refers to the study of vectorspaces, where a vectorspace is a collection of "things" and a kind of "arithmetic" that can be done with those things.  This is made precise by the definition of vectorspace, and includes "multiplying" the things by numbers (elements of a fixed field).  The things in the vectorspace are called vectors, but can be very different from the vectors of physics.  For example they might be polynomials, or matrices, or certain functions.  The simplest types of vector spaces are the n-tuples of real numbers which make up n-dimensional Euclidean space, where n is a fixed (but arbitrary) positive integer.