Linear Algebra (Honours)

Instructor: Yaniv Plan
Office: 1219 Math Annex
Email: yaniv (at) math (dot) ubc (dot) ca

Lectures: MWF 10:00-11:00am, MATX 1100.

Office hours:

  • MW 3:30pm-4:30pm, MATX 1219, Th 5:00pm-6:00pm, MATX 1118

Course outline, grading scheme, etc.: Outline.

Final: Thursday, December 5, 8:30-11, in MCLD 202.

  • Cumulative. Covers all material aside from lecture on my research, the climate strike lecture on temperature prediction, and analysis of accuracy of least squares.
  • Will be challenging like midterm, but with more time per problem
  • Sample Final which best fits our material
  • You may also study previous final exams
    • 2013 Final (aside from problem 7). Notation: P_3 is the set of degree-3 polynomials. Sym_2 is symmetric 2 by 2 matrices.
    • 2012 Final (all problems).
  • Algorithmic problems/topics that you should know:
    • Compute determinant
    • Gaussian elimination
    • Matrix diagonalization, orthogonal diagonalization for symmetric matrices
    • Find a basis for R(A) (using columns of A), and a basis for N(A) given A, or given the singular vectors of A.
    • Solve linear recurrence relations (e.g., Fibonacci numbers)
    • Solve linear system of differential equations
    • Gram Schmidt (including vector spaces aside from R^n, C^n)
    • Least squares

Midterm: Wednesday, October 23, in class.

  • Sample Midterm 1, Sample Midterm 2 (for 4b, we say matrices are similar if they have the same eigen-values). I won’t post solutions but you can ask about solutions in office hours.
  • Closed book, no notes, not calculators
  • Covers everything through the notes on Change of Basis
  • A proportion of the test will cover the more straightforward algorithmic problems. For this proportion of the test, be sure you understand/can do the following:
    • Compute determinant
    • Gaussian elimination
    • Matrix diagonalization
    • Find a basis for R(A) (using columns of A), and a basis for N(A)
    • Solve linear recurrence relations (e.g., Fibonacci numbers)

Math Learning Centre.

(MLC for short) A space for undergraduate students to study math together, with friendly support from tutors, who are graduate and undergraduate students in the math department. The MLC is located at LSK301 and LSK302. Every undergraduate student studying Math is welcome there! In the MLC, students may join the study groups if students wish to. Please note that while students are encouraged to seek help with homework, the MLC is not a place to check answers or receive solutions, rather, the aim is to aid students in becoming better learners and to develop critical thinking in a mathematical setting. For additional information please visit the website.

Notes. Written by Richard Anstee. These give an intuitive approach to our material.

2×2 matrices warm up. Matrix algebra.

2×2 matrices determinants and inverses.

Putnam problem about integer coefficients.

2×2 matrices are linear transformations are 2×2 matrices.

Different levels of generality. (Not discussed in class. Read me!)

Eigenvectors and eigenvalues (long run behaviour of bird populations).

Fibonacci numbers.

Change of coordinates.

Gaussian elimination.

Determinants.

Interchanging rows changes sign of determinant. (Technical part of proof from “Determinants”. Not discussed in class.)

Partial Fractions. (Not covered.)

Vector space, field, axioms. (Abstract definition of vector space. Missing one axiom: For every v V, there exists an element v V, called the additive inverse of v, such that v + ( v) = 0)

Intuitive intro to abstract vector space. Subspace. Span.

Linear independence, spanning set, dimension.

Row space, column space, and rank.

Change of basis.

Systems of differential equations.

Complex numbers 1 and Complex numbers 2.

Vector geometry. Dot product, length, angle in high dimensions.

Orthogonal vector spaces.

Gram Schmidt algorithm, orthonormal basis, orthonormal matrix.

Orthogonal projections and least squares.

Symmetric matrices are orthogonally diagonalizable.

Singular value decomposition (SVD)
Lecture from 11/22. Scribe: Nicole.

Lecture from 11/25. Scribe: Nicole.

Lecture from 11/27. Scribe: Nicole.

Reference for singular value decomposition (SVD): Linear Algebra and its Applications, by D. Lay, Editions 4 or 5 (but not 3). You can find it at the library. Here is a pdf of the relevant chapter: SVD.

Homework.

Assignment 1. Due Friday, September 13, at beginning of class. Solutions.

Assignment 2. Due Friday, September 20, at beginning of class. Solutions.

Assignment 3. Due Friday, October 4, at the beginning of class. Solutions.
Note: You have two weeks for this one, but it’s not twice as long! For students who are rewriting parts of the first assignment, please take advantage. Also, please practice Gaussian elimination until you are comfortable with it. (See Linear Algebra and its Applications, by D. Lay, for plenty of examples.)

Assignment 4. Due Friday, October 11, at the beginning of class. Solutions.

Assignment 5. Due Friday, October 18, at the beginning of class. Solutions.

Assignment 6. Due Friday, November 1, at the beginning of class. Solutions.

Assignment 7. Due Friday, November 8, at the beginning of class. Solutions.

Here is the mark sheet for the first 5 homeworks and midterm. It contains student numbers but not names for sake of anonymity. Please double-check for correctness.

Assignment 8. Due Friday, November 22. (That’s two weeks, enjoy a break on the long weekend!) Solutions.

  • A chance for another challenging problem: Starting on November 13, come to my office hours and I will find a challenging question for you, perhaps related to my research. Not for credit.
  • I’ll be away November 15 (Prof. Robeva will teach the class that day).
  • My office hour on November 14 is canceled.

Assignment 9. Due Friday, November 29.  Solutions.

Here is the mark sheet for all of the homeworks.  Please double-check for correctness and report any error by Friday, December 6, at the latest.