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). The matrix of a linear transformation.
Projections, reflections, rotations, and their matrices. Orthogonal projection in the plane.
Matrix multiplication. Reading: 2.1, 2.2. Homework 3.
Week 4. Quiz 3 (linear transformations). Matrix multiplication and inversion.
Reading: 2.3, 2.4. Homework 4. Midterm 1 (1.1-1.3, 2.1-2.4).
Week 5. Matrix inversion. LU example. Kernel and image of a matrix. Span of vectors.
Quiz 4 (matrix inversion). Subspaces of Euclidean space. Redundancy and linear independence.
Week 6. Characterizations of linear (in)dependence. Basis and dimension.
Week 7. Coordinates. Matrix of a linear transformation with respect to a given basis. Case of the plane.
Quiz 6 (basis and dimension). Transpose. Orthonormal bases. Orthogonal projection onto a subspace.
Week 8. Orthogonalization of vectors. Orthogonal matrices.
Reading: 5.2, 5.3. Homework 8. Midterm 2 (3.1-3.4, 5.1-5.3).
Week 9. Least squares solutions. Determinants. Notes on determinants.
Cramer’s rule. Eigenvalues and eigenvectors. Examples.
Week 10. Quiz 7 (eigenvalues and eigenvectors). Characteristic polynomial. Eigenspaces. Algebraic and geometric multiplicity.
Reading: 7.2, 7.3. Homework 10.
June 11. Office hours: 2-3 (Library Terrace).
June 12. Final exam: 10:30-12:30, Disque 108.