# Math 201 – Linear Algebra, Spring 2018

Instructor: Anatolii Grinshpan
Office hours:  Mon 2-3 and Wed 2-4, Library Terrace.

Week 1. Introduction to the course. Gaussian elimination. Leading and free variables.

Reduced echelon form. Rank. Example. Tall, long, square.

Reading: 1.1, 1.2. Homework 1. Uniqueness of RREF.

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.. 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.

Quiz 5 (kernel and image). Basis for the kernel Rank-nullity theorem.

Reading: 3.3.  Homework 6. Notes on bases.

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.

Reading: 3.4, 5.1. Homework 7a. Homework 7b. Homogeneous equations and orthogonality.

Week 8. Orthogonalization of vectors. Orthogonal matrices.

Reading: 5.2, 5.3. Midterm 2 (3.1-3.4, 5.1-5.3).

Week 9.  Least squares solutionsDeterminants.

Cramer’s rule. Eigenvalues and eigenvectors. Examples.

Reading: 6.1-6.3, 7.1. Homework 9a. Homework 9b.

Week 10. Quiz 7 (eigenvalues and eigenvectors). Characteristic polynomial. Eigenspaces. Algebraic and geometric multiplicity.

Eigenbases and diagonalization.  Examples. Cross-product transformation.