The coverage is standard: linear systems and Gauss' method, vector spaces, linear maps and matrices, determinants, and eigenvectors and eigenvalues. Prerequisites: A semester of calculus. Students with three semesters of calculus can skip a few sections. Applications: Each chapter has three or four discussions of additional topics and applications. These are suitable for independent study or for small group work. What makes it different? The approach is developmental. Although the presentation is focused on covering the requisite material by proving things, it does not start with an assumption that students are already able at abstract work. Instead, it proceeds with a great deal of motivation, many computational examples, and exercises that range from routine verifications to (a few) challenges. The goal is, in the context of developing the usual material of an undergraduate linear algebra course, to help raise the level of mathematical maturity of the class.Linear Systems.
Solving Linear Systems.
Gauss’ Method.
Describing the Solution Set.
General = Particular + Homogeneous.
Linear Geometry of n-Space.
Vectors in Space.
Length and Angle Measures.
Reduced Echelon Form.
Gauss-Jordan Reduction.
Row Equivalence.
Vector Spaces.
Definition of Vector Space.
Definition and Examples.
Subspaces and Spanning Sets.
Linear Independence.
Definition and Examples.
Basis and Dimension.
Basis.
Dimension.
Vector Spaces and Linear Systems.
Combining Subspaces.
Maps Between Spaces.
Isomorphisms.
Definition and Examples.
Dimension Characterizes Isomorphism.
Homomorphisms.
Definition.
Rangespace and Nullspace.
Computing Linear Maps.
Representing Linear Maps with Matrices.
Any Matrix Represents a Linear Map.
Matrix Operations.
Sums and Scalar Products.
Matrix Multiplication.
Mechanics of Matrix Multiplication.
Inverses.
Change of Basis.
Changing Representations of Vectors.
Changing Map Representations.
Projection.
Orthogonal Projection Into a Line.
Gram-Schmidt Orthogonalization.
Projection Into a Subspace.
Determinants.
Definition.
Exploration.
Properties of Determinants.
The Permutation Expansion.
Determinants Exist.
Geometry of Determinants.
Determinants as Size Functions.
Other Formulas.
Laplace’s Expansion.
Similarity.
Complex Vector Spaces.
Factoring and Complex Number.
Complex Representations.
Similarity.
Definition and Examples.
Diagonalizability.
Eigenvalues and Eigenvectors.
Nilpotence.
Self-Composition.
Strings.
Jordan Form.
Polynomials of Maps and Matrices.
Jordan Canonical Form.
