Math 56 - Linear Algebra -- [4 units]
Course Format: Three hours of lecture and three hours of discussion per week.Prerequisites: Basic algebra and trigonometry. Familiarity with logical arguments and mathematical proofs.
Credit Restrictions: Students will receive no credit for MATH 56 after completing MATH 54, MATH N54, or MATH W54
Description: This is a first course in Linear Algebra. Core topics include: algebra and geometry of vectors and matrices; systems of linear equations and Gaussian elimination; eigenvalues and eigenvectors; Gram-Schmidt and least squares; symmetric matrices and quadratic forms; singular value decomposition and other factorizations. Time permitting, additional topics may include: Markov chains and Perron-Frobenius, dimensionality reduction, or linear programming. This course differs from Math 54 in that it does not cover Differential Equations, but focuses on Linear Algebra motivated by first applications in Data Science and Statistics.
Textbook: Gupta-Nadler-Paulin, Linear Algebra, available online at: https://drive.google.com/file/d/1qPBRIdTosqIVS2-GLO75Xojm-IrnyVku/view
| Course Outline | |
| Chapter 1: Vectors Algebra and geometry of vectors, linear independence and spanning sets, subspaces, bases and dimension. (Sections 1.1–1.5.) |
6 hours |
| Chapter 2: Linear functions and linear transformations Multivariable functions, linear transformations as weighted networks, linear equations and matrix equations, fundamental subspaces and preimages. (Sections 2.1–2.3.) |
4.5 hours |
| Chapter 3: Solving matrix equations Solutions in parametric form, reduced row echelon form, Gaussian elimination. (Sections 3.1–3.2.) |
3 hours |
| Chapter 4: Applications to vectors and linear transformations Existence and uniqueness properties, computations of bases and dimensions. (Sections 4.1–4.2.) |
3 hours |
| Chapter 5-6: Matrix algebra Matrix addition and multiplication, composition of linear transformations, algebra of row operations, relation of matrix transpose and dot product, invertible and triangular matrices. (Sections 5.1–5.2, 6.1–6.2.) |
3 hours |
| Chapter 7: Determinants Area and volume, geometry of determinants, algebra of determinants, permutation matrices. (Sections 7.1–7.3.) |
1.5 hours |
| Chapter 8-9: Orthogonal bases and projections Orthogonal bases and matrices, Gram-Schmidt and QR factorization, distance to subspace, least-squares solutions. (Sections 8.1-8.2, 9.1-9.2) |
4 hours |
| Chapter 10-11: Coordinates and simplifying matrices Coordinates on subspaces, changes of coordinates, from linear transformations to matrices, simplifying matrices, similarity. (Sections 10.1–10.3, 11.1-11.2.) |
3 hours |
| Chapter 12: Eigenvalues and eigenvectors Characteristic polynomial, Fundamental Theorem of Algebra and complex eigenvalues, eigenvectors and eigenspaces, diagonalizability. |
4 hours |
|
Chapter 13: Spectral Theorem and Singular Value Decomposition
Symmetric matrices and quadratic forms, orthogonal diagonalizability, algebra and geometry of singular value decompositions, low rank approximations. (Sections 13.1–13.2.)
|
4 hours |
| Selected Applications | 2 hours |
| Total | 38 hours |
| Midterms & Holidays | 4 hours |