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 edition) and Nagle-Saff-Snider, Fundamentals of Differential Equations and Boundary Value Problems (9th edition). A specially priced UC Berkeley paperback edition is availablecontaining 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