Math 54

Math 54 - Linear Algebra & Differential Equations -- [4 units]

Course Format: Three hours of lecture and three hours of discussion per week.

Prerequisites: 1A-1B, 10A-10B or equivalent. 

Description: Basic linear algebra; matrix arithmetic and determinants. Vector spaces; inner product spaces. Eigenvalues and eigenvectors; linear transformations, symmetric matrices. Linear ordinary differential equations (ODE); systems of linear ODE.  Fourier series. (F,SP)

Textbook: Lay-Lay-McDonald, Linear Algebra and its Applications (5th and 6th editions) and Nagle-Saff-Snider, Fundamentals of Differential Equations and Boundary Value Problems (9th edition).A specially priced UC Berkeley paperback version (2nd edition) is available containing the chapters of both books needed for the course.

Part One: (Lay et al.)
Chapter 1: Linear Equations in linear algebra
Systems of linear equations, row reduction, vectors in Rn, linear independence, matrices, linear transformations. (Sections 1.1–1.5, 1.7–1.9.)
5 hours
Chapter 2-3: Matrix Algebra and Determinants
Matrix algebra, the inverse of a matrix, Determinants: properties, computation by row reduction or cofactor expansion, geometric interpretation in terms of volume. (Sections 2.1–2.3, 3.1–3.3.)
4 hours
Chapter 4: Vector Spaces
Vector spaces and subspaces, including examples of function spaces, nullspace (kernel) and column space (image) of a matrix (linear transformation), bases, coordinate systems, dimension and rank, change of basis. (Sections 4.1–4.7)
6 hours
Chapter 5: Eigenvalues and Eigenvectors
Eigenvalues and eigenvectors, the characteristic equation, diagonalization, eigenvectors and linear transformations, complex eigenvalues. (Sections 5.1–5.5 and Appendix B.)
4 hours
Chapter 6: Orthogonality, Least Squares
The Euclidean inner product on Rn, orthogonal sets, orthogonal projection, Gram-Schmidt process, least squares problems, applications to linear models, inner product spaces. (Sections 6.1–6.7.)
6 hours
Chapter 7: Symmetric Matrices, Applications
Diagonalization of symmetric matrices, quadratic forms, singular value decomposition. (Sections 7.1–7.2, 7.4.)
3 hours
Total hours on Linear Algebra 28 hours
Part Two: (Nagle et al.)
Chapter 4: Linear Second-Order ODE
Linear second-order ODE: homogeneous equations, inhomogeneous equations using the method of undetermined coefficients. (Sections 4.2–4.5.)
3 hours
Chapter 9: Systems of linear ODE
Systems of first order linear ODE: reduction of higher order equations to single order systems, homogeneous constant coefficient equations using eigenvalues (Sections 9.1, 9.4–9.6)
4 hours
Chapter 10: Fourier Series
Fourier series. (Sections 10.3, 10.4)
3 hours
Total hours on Differential Equations 10 hours
Total for Parts One and Two 38 hours
Midterms & holidays  4 hours
The total number of class hours may vary during the Fall and Spring semesters. 42 hours