0.2: Suggested Course Outlines

Chapter 1: Systems of Linear Equations.

A standard treatment of gaussian elimination is given. The rank of a matrix is introduced via the row-echelon form, and solutions to a homogeneous system are presented as linear combinations of basic solutions. Applications to network flows, electrical networks, and chemical reactions are provided.

Chapter 2: Matrix Algebra.

After a traditional look at matrix addition, scalar multiplication, and transposition in Section [sec:2_1], matrix-vector multiplication is introduced in Section [sec:2_2] by viewing the left side of a system of linear equations as the product \(A\vect{x}\) of the coefficient matrix \(A\) with the column \(\vect{x}\) of variables. The usual dot-product definition of a matrix-vector multiplication follows. Section [sec:2_2] ends by viewing an \(m \times n\) matrix \(A\) as a transformation \(\RR^n \to \RR^m\). This is illustrated for \(\RR^2 \to \RR^2\) by describing reflection in the \(x\) axis, rotation of \(\RR^2\) through \(\frac{\pi}{2}\), shears, and so on.

In Section [sec:2_3], the product of matrices \(A\) and \(B\) is defined by \(AB = \leftB \begin{array}{cccc} A\vect{b}_1 & A\vect{b}_2 & \cdots & A\vect{b}_n \end{array}\rightB\), where the \(\vect{b}_i\) are the columns of \(B\). A routine computation shows that this is the matrix of the transformation \(B\) followed by \(A\). This observation is used frequently throughout the book, and leads to simple, conceptual proofs of the basic axioms of matrix algebra. Note that linearity is not required—all that is needed is some basic properties of matrix-vector multiplication developed in Section [sec:2_2]. Thus the usual arcane definition of matrix multiplication is split into two well motivated parts, each an important aspect of matrix algebra. Of course, this has the pedagogical advantage that the conceptual power of geometry can be invoked to illuminate and clarify algebraic techniques and definitions.

In Section [sec:2_4] and [sec:2_5] matrix inverses are characterized, their geometrical meaning is explored, and block multiplication is introduced, emphasizing those cases needed later in the book. Elementary matrices are discussed, and the Smith normal form is derived. Then in Section [sec:2_6], linear transformations \(\RR^n \to \RR^m\) are defined and shown to be matrix transformations. The matrices of reflections, rotations, and projections in the plane are determined. Finally, matrix multiplication is related to directed graphs, matrix LU-factorization is introduced, and applications to economic models and Markov chains are presented.

Chapter 3: Determinants and Diagonalization.

The cofactor expansion is stated (proved by induction later) and used to define determinants inductively and to deduce the basic rules. The product and adjugate theorems are proved. Then the diagonalization algorithm is presented (motivated by an example about the possible extinction of a species of birds). As requested by our Engineering Faculty, this is done earlier than in most texts because it requires only determinants and matrix inverses, avoiding any need for subspaces, independence and dimension. Eigenvectors of a \(2 \times 2\) matrix \(A\) are described geometrically (using the \(A\)-invariance of lines through the origin). Diagonalization is then used to study discrete linear dynamical systems and to discuss applications to linear recurrences and systems of differential equations. A brief discussion of Google PageRank is included.

Chapter 4: Vector Geometry.

Vectors are presented intrinsically in terms of length and direction, and are related to matrices via coordinates. Then vector operations are defined using matrices and shown to be the same as the corresponding intrinsic definitions. Next, dot products and projections are introduced to solve problems about lines and planes. This leads to the cross product. Then matrix transformations are introduced in \(\RR^3\), matrices of projections and reflections are derived, and areas and volumes are computed using determinants. The chapter closes with an application to computer graphics.

Chapter 5: The Vector Space \(\RR^n\).

Subspaces, spanning, independence, and dimensions are introduced in the context of \(\RR^n\) in the first two sections. Orthogonal bases are introduced and used to derive the expansion theorem. The basic properties of rank are presented and used to justify the definition given in Section [sec:1_2]. Then, after a rigorous study of diagonalization, best approximation and least squares are discussed. The chapter closes with an application to correlation and variance.

This is a “bridging” chapter, easing the transition to abstract spaces. Concern about duplication with Chapter [chap:6] is mitigated by the fact that this is the most difficult part of the course and many students welcome a repeat discussion of concepts like independence and spanning, albeit in the abstract setting. In a different direction, Chapter [chap:1]–[chap:5] could serve as a solid introduction to linear algebra for students not requiring abstract theory.

Chapter 6: Vector Spaces.

Building on the work on \(\RR^n\) in Chapter [chap:5], the basic theory of abstract finite dimensional vector spaces is developed emphasizing new examples like matrices, polynomials and functions. This is the first acquaintance most students have had with an abstract system, so not having to deal with spanning, independence and dimension in the general context eases the transition to abstract thinking. Applications to polynomials and to differential equations are included.

Chapter 7: Linear Transformations.

General linear transformations are introduced, motivated by many examples from geometry, matrix theory, and calculus. Then kernels and images are defined, the dimension theorem is proved, and isomorphisms are discussed. The chapter ends with an application to linear recurrences. A proof is included that the order of a differential equation (with constant coefficients) equals the dimension of the space of solutions.

Chapter 8: Orthogonality.

The study of orthogonality in \(\RR^n\), begun in Chapter [chap:5], is continued. Orthogonal complements and projections are defined and used to study orthogonal diagonalization. This leads to the principal axes theorem, the Cholesky factorization of a positive definite matrix, QR-factorization, and to a discussion of the singular value decomposition, the polar form, and the pseudoinverse. The theory is extended to \(\mathbb{C}^n\) in Section [sec:8_6] where hermitian and unitary matrices are discussed, culminating in Schur’s theorem and the spectral theorem. A short proof of the Cayley-Hamilton theorem is also presented. In Section [sec:8_7] the field \(\mathbb{Z}_p\) of integers modulo \(p\) is constructed informally for any prime \(p\), and codes are discussed over any finite field. The chapter concludes with applications to quadratic forms, constrained optimization, and statistical principal component analysis.

Chapter 9: Change of Basis.

The matrix of general linear transformation is defined and studied. In the case of an operator, the relationship between basis changes and similarity is revealed. This is illustrated by computing the matrix of a rotation about a line through the origin in \(\RR^3\). Finally, invariant subspaces and direct sums are introduced, related to similarity, and (as an example) used to show that every involution is similar to a diagonal matrix with diagonal entries \(\pm 1\).

Chapter 10: Inner Product Spaces.

General inner products are introduced and distance, norms, and the Cauchy-Schwarz inequality are discussed. The Gram-Schmidt algorithm is presented, projections are defined and the approximation theorem is proved (with an application to Fourier approximation). Finally, isometries are characterized, and distance preserving operators are shown to be composites of a translations and isometries.

Chapter 11: Canonical Forms.

The work in Chapter [chap:9] is continued. Invariant subspaces and direct sums are used to derive the block triangular form. That, in turn, is used to give a compact proof of the Jordan canonical form. Of course the level is higher.