# 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 algebraSystems 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 DeterminantsMatrix 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 SpacesVector 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 EigenvectorsEigenvalues 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 SquaresThe 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, ApplicationsDiagonalization 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 ODELinear 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 ODESystems 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 SeriesFourier 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
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