Theory and problems of linear algebra - Seymour Lipschutz (McGraw-Hill, 1991)
The first chapter of book treats systems of linear equations. This provides the motivation and basic computational tools for the subsequent material. After vectors and matrices are introduced, there are chapters on vector spaces and subspaces and on inner products. This is followed by chapters covering determinant and on eigenvalues and eigenvectors, and diagonalizing matrices and quadratic forms. The later chapters cover abstract linear maps and their canonical forms, specifically the triangular, Jordan and rational canonical forms. The last chapter treats abstract linear maps on inner products spaces