By using linear algebra seemingly different "linear" things in algebra, geometry, and calculus (things like systems of certain equations, rigid motions in geometry, certain differential equations) are placed in a common framework of vectors, matrices, and linear transformations.
Viewing different things as somehow analogous provides not only insight, but also a common toolkit of surprisingly powerful algorithms.
This book introduces you in the theory of vector spaces, but also to the practical considerations of using a computer to carry out such algorithms.