A unique treatment of linear algebra which establishes the central role of invariant subspaces in the analysis of linear transformation. Incorporating the newest developments in linear algebra stimulated by linear systems theory, it gives a comprehensive view of geometrical, algebraic, topological, and analytical properties of invariant subspaces, with an emphasis on applications to matrix polynomials, rational matrix functions, linear systems, and matrix quadratic equations. Presents an algebraic treatment of control and systems theories. Written by a world-famous expert, this text contains material not previously published. Includes numerous exercises.