ENGLISH

Linear Algebra, Multivariable Calculus, and Modern Applications

Book information

Publisher
Stanford University
Year
2021
Language
english
Format
PDF
Filesize
11 MB (11683079 bytes)
Pages
\729
Time added
2022-05-20 03:42:46

Description

Math 51 course text prepared by the Stanford University Math Department Last modified on March 10, 2021 Introduction Applications Advice on studying, homework, and exams for math in college, and tutoring/online resources Preparedness for Math 51 Advice to Instructors Part I. Geometry of vectors and projections 1. Vectors, vector addition, and scalar multiplication 2. Vector geometry in Rn and correlation coefficients 3. Planes in R3 4. Span, subspaces, and dimension 5. Basis and orthogonality 6. Projections 7. Applications of projections in Rn: orthogonal bases of planes and linear regression Part II. Multivariable functions and optimization 8. Multivariable functions, level sets, and contour plots 9. Partial derivatives and contour plots 10. Maxima, minima, and critical points 11. Gradients, local approximations, and gradient descent 12. Constrained optimization via Lagrange multipliers Part III. Geometry and algebra of matrices 13. Linear functions, matrices, and the derivative matrix 14. Linear transformations and matrix multiplication 15. Matrix algebra 16. Applications of matrix algebra: population dynamics, PageRank, and gambling 17. Multivariable Chain Rule 18. Matrix inverses and multivariable Newton's method for zeros Part IV. Further matrix algebra and linear systems 19. Linear independence and the Gram–Schmidt process 20. Matrix transpose, quadratic forms, and orthogonal matrices 21. Linear systems, column space, and null space 22. Matrix decompositions: QR-decomposition and LU-decomposition Part V. Eigenvalues and second partial derivatives 23. Eigenvalues and eigenvectors 24. Applications of eigenvalues: Spectral Theorem, quadratic forms, and matrix powers 25. The Hessian and quadratic approximation 26. Grand finale: application of the Hessian to local extrema, and bon voyage 27. More eigenvalue applications: ODE systems, population dynamics, SVD (optional) Appendices A. Review of functions B. Further details on linear algebra results (optional) C. Equivalence of two perspectives on ellipses and hyperbolas (optional) D. Google's PageRank algorithm (optional) E. General determinants (optional) F. The cross product (optional) G. Neural networks and the multivariable Chain Rule (optional) H. The QR algorithm (optional) I. Newton's method for optimization (optional) J. Hessians and chemistry (optional) References

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