Linear Algebra (Honours)

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

Lectures: MWF 10:00-11:00am, 460 CSCI.

Office hours:

  • Wednesdays 4:00-5:00pm, 1219 Math Annex.
  • Tuesdays 4:00-5:00pm, 105 Math.
  • FINAL REVIEW SESSIONS:  Mondays 3-5pm.  (No review session on April 17 since it is a holiday.  However, I will be around the office most of the afternoon on Wednesday and you are welcome to stop by.)

Course outline, grading scheme, etc.: Outline.

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.

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

Intuitive intro to abstract vector space.  Subspace.  Span.  (Midterm covers material through this lecture.)

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.

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, January 20, at the beginning of class.  Solutions.

Assignment 2.  Due Friday, January 27, at the beginning of class.  Solutions.

Assignment 3.  Due Friday, February 3, at the beginning of class.  Solutions.

Assignment 4.  Due Friday, February 10, at the beginning of class.  Solutions.

Assignment 5.  Deadline extended!  Due Monday, March 6, at the beginning of class.  (But future homeworks will still be due on Fridays!)  Solutions.

Assignment 6.  Due Friday, March 10, at the beginning of class.  Solutions.

Assignment 7.  Deadline extended! Due Monday, March 20, at the beginning of class.  Solutions.

Assignment 8.  Due Monday!  March 27, at the beginning of class.  Solutions.

Assignment 9.  Due Wednesday, April 5, at the beginning of class.  Solutions.

 

Midterm:  February 17 in class.  You may use a 3 inch by 5 inch index card of notes (both sides), writing utensils, but nothing else (no calculators).  You can also cut out a piece of paper that is 3 inches by 5 inches.

Sample midterm.  I will not post solutions, but you are welcome to ask about solutions in office hours.  Note: This sample midterm does not have any abstract problems or proofs.  However, our midterm may have a few abstract problems and proofs.  There is a decent chance it will have a proof by induction.  Possibly also a proof using only the axioms of a field or vector space.

Sample Midterm 1.

Final exam.  April 18 at 8:30am.  Location: LASR 102.  It will cover all of the material from class, with a small emphasis on the material that the midterm did not cover.  You may use a 3 inch by 5 inch index card of notes (both sides), writing utensils, but nothing else (no calculators).  You can also cut out a piece of paper that is 3 inches by 5 inches.  Note:  Singular value decomposition is likely to appear on the final.

2008 Sample Final exam.
2009 Sample Final exam.

I will not post solutions, but you are welcome to ask about solutions in office hours.