# 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 5 hours Systems of linear equations, row reduction, vectors in $\mathbb R^n$, linear independence, matrices, linear transformations. (Sections 1.1–1.5, 1.7–1.9.) Chapter 2-3: Matrix Algebra and Determinants 4 hours 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.) Chapter 4: Vector Spaces 6 hours 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) Chapter 5: Eigenvalues and Eigenvectors 4 hours Eigenvalues and eigenvectors, the characteristic equation, diagonalization, eigenvectors and linear transformations, complex eigenvalues. (Sections 5.1–5.5 and Appendix B.) Chapter 6: Orthogonality, Least Squares 6 hours The Euclidean inner product on $\mathbb R^n$, orthogonal sets, orthogonal projection, Gram-Schmidt process, least squares problems, applications to linear models, inner product spaces. (Sections 6.1–6.7.) Chapter 7: Symmetric Matrices, Applications 3 hours Diagonalization of symmetric matrices, quadratic forms, singular value decomposition. (Sections 7.1–7.2, 7.4.) Total hours on Linear Algebra 28 hours Part Two: (Nagle et al.) Chapter 4: Linear Second-Order ODE 3 hours Linear second-order ODE: homogeneous equations, inhomogeneous equations using the method of undetermined coefficients. (Sections 4.2–4.5.) Chapter 9: Systems of linear ODE 4 hours 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) Chapter 10: Fourier Series 3 hours Fourier series. (Sections 10.3, 10.4) 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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