Preface The title of the book sounds a bit mysterious. Why should anyone read this book if it presents the subject in a wrong way? What is particularly done " wrong " in the book? Before answering these questions, let me first describe the target audience of this text. This book appeared as lecture notes for the course " Honors Linear Algebra ". It supposed to be a first linear algebra course f...

We investigate similarities between the category of vector spaces and that of polytopal algebras, containing the former as a full subcategory. In Section 2 we introduce the notion of a polytopal Picard group and show that it is trivial for fields. The coincidence of this group with the ordinary Picard group for general rings remains an open question. In Section 3 we survey some of the previous ...

Two real entried (respectively: orthogonal) n x n matrices^! andB are said to be linearly similar if there is an invertible real (resp. orthogonal) n x n matrix C with CAC~ = B. Of course, A, B and C may be regarded as linear functions from R (resp. isometries from S~) to itself. The matrices A and B are said to be (resp. homogeneously differentiably) differentiably similar if there is a diffeo...

3 The Exterior Algebra 6 3.1 Free modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.3 The trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.4 Interpolating between trace and determinant . . . . . . . . . . . . . . . . ...

1 Basics 2 Linear Equations 3 Linear Maps 4 Rank One Matrices 5 Algebra of Matrices 6 Eigenvalues and Eigenvectors 7 Inner Products and Quadratic Forms 8 Norms and Metrics 9 Projections and Reflections 10 Similar Matrices 11 Symmetric and Self-adjoint Maps 12 Orthogonal and Unitary Maps 13 Normal Matrices 14 Symplectic Maps 15 Differential Equations 16 Least Squares 17 Markov Chains 18 The Expo...

The pivotal problem in computational linear algebra is how to balance data access costs with the cost of computation. Given the disparity between memory and CPU speeds, it is critical to maximize the amount of computation carried out for each item of data accessed from memory. For problems for which the systems of equations associated with a problem are dense (i.e. they contain mostly non-zero ...