## 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 R ^{n}, 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 R ^{n}, 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 |

**Documents:**