A Vector Space, as a purely abstract concept, about which we prove properties and theorems, is part of Abstract Algebra, and well integrated into this field. Some striking examples of this are the group of invertible linear maps or Matrices, and the ring of linear maps of a vector space. On the other side of Mathematics, Linear Algebra plays an important part in Analysis, notably, in the study of Tensor Products and Alternative Maps. But, to begin at the beginning, one has to define some elementary objects and properties on which Linear Algebra is built and look at some examples. Linear Algebra has to do with Matrices .
Matrices are basically tables of numbers.
Surprisingly enough, studying their properties can be very enlightening. Linear Algebra covers such arcane topics as Eigenvectors .