Math 54
Math 54  Linear Algebra & Differential Equations  [4 units]
Course Format: Three hours of lecture and three hours of discussion per week.
Prerequisites: 1A1B, 10A10B 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: LayLayMcDonald, Linear Algebra and its Applications (5th and 6th editions) and NagleSaffSnider, 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 23: 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, GramSchmidt 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 SecondOrder ODE 
3 hours 
Linear secondorder 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 