Week 1. Introduction to the course. Gaussian elimination. Leading and free variables.
Week 2. Quiz 1 (Gauss-Jordan elimination). Vector arithmetic. Matrix-vector product.
Homogeneous and nonhomogeneous systems: the structure of solutions.
Matrices as linear transformations. Reading: 1.3. Homework 2.
Week 3. Quiz 2 (matrix-vector product). Linear transformations. The matrix of a linear transformation.
Projections, reflections, rotations and their matrices. Orthogonal projection in the plane.
Week 4. Matrix multiplication. Elementary matrices. Dihedral group. Quiz 3 (linear transformations).
One-to-one and onto linear transformations. Matrix inversion. LU example.
Week 5. Midterm 1 (weeks 1-4). Block multiplication and inversion. Kernel and image of a matrix.
Span of vectors. Linear dependence and independence.
Week 6. Subspaces associated to a matrix. Quiz 4 (kernel and image). Characterizations of independence.
Basis and dimension. Basis for the kernel and basis for the image. Rank-nullity theorem.
Week 7. Coordinates. Matrix of a linear transformation with respect to a given basis.
Examples. Transpose. Orthonormal bases.
Reading: 3.3, 3.4, 5.1. Homework 10.
Week 8. Quiz 5 (matrix in a given basis). Orthogonal complement. Orthogonal projection onto a subspace.
Week 9. Midterm 2 (weeks 5-8). Thanksgiving break.
Week 10. Least squares solutions. Determinants. Cramer’s rule.
Eigenvalues and eigenvectors. Examples.
Week 11. Quiz 6 (determinants). Eigenspaces and eigenbases. Diagonalization.
Dec 13. Office hours: 1-3 (Library Terrace).
Dec 14. Questions session: 12-2, Disque 108.
Dec 15. Final exam: 10:30-12:30, Stratton 113.