Linear Algebra, Multivariable Calculus, and Modern Applications

• Publisher: Stanford University
• Year: 2021
• Tongue: English
• Leaves: 729
• Category: Library

✦ Synopsis

Math 51 course text prepared by the Stanford University Math Department
Last modified on March 10, 2021

✦ Table of Contents

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