Linear Algebra : Introducction

It is all about generalizing the concepts of space and dimension.
Matrix algebra involves transformations to appropriate forms by sometimes changing the basis.
EigneValues are a very good example where the Vectors are first converted to the basis of eignevectors. The calculations with the transormation is easier since A*x=LAMBDA*x for all eignevectors. After calculations are done , the transformation is again done to the original basis.