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.
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Part One: (Lay et al.) |
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Chapter 1: Linear Equations in linear algebra |
5 hours |
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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.) |
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Chapter 2-3: Matrix Algebra and Determinants |
4 hours |
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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.) |
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Chapter 4: Vector Spaces |
6 hours |
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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) |
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Chapter 5: Eigenvalues and Eigenvectors |
4 hours |
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Eigenvalues and eigenvectors, the characteristic equation, diagonalization, eigenvectors and linear transformations, complex eigenvalues. (Sections 5.1–5.5 and Appendix B.) |
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Chapter 6: Orthogonality, Least Squares |
6 hours |
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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.) |
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Chapter 7: Symmetric Matrices, Applications |
3 hours |
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Diagonalization of symmetric matrices, quadratic forms, singular value decomposition. (Sections 7.1–7.2, 7.4.) |
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Total hours on Linear Algebra |
28 hours |
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Part Two: (Nagle et al.) |
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Chapter 4: Linear Second-Order ODE |
3 hours |
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Linear second-order ODE: homogeneous equations, inhomogeneous equations using the method of undetermined coefficients. (Sections 4.2–4.5.) |
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Chapter 9: Systems of linear ODE |
4 hours |
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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) |
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Chapter 10: Fourier Series |
3 hours |
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Fourier series. (Sections 10.3, 10.4) |
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Total hours on Differential Equations |
10 hours |
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Total for Parts One and Two |
38 hours |
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Midterms + holidays |
4 hours |
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The total number of class hours may vary during the Fall and Spring semesters. |
42 hours |
