MAT240F Algebra I

Classes: T11-1, R12; Room: HS610. (Health Sciences, 155 College Street).

Midterm exam: T11-1 on October 17, 2017 , in EX 100

 Course description

This course is an introduction to Linear Algebra, aimed at students in our specialist programs.

Text Book: Linear Algebra (fourth edition) by Friedberg, Insel and Spence, Prentice Hall. (Required reading.)

Prerequisite: High school level calculus

Corequisite: MAT157Y

Course outline

Course syllabus

The required homework problems for this course will be sent online, via Crowdmark. You will be asked to submit the solutions electronically, via Crowdmark. (More information will be given later.)

Our late policy is as follows: No late assignments will be accepted.

You can certainly discuss homework with classmates, but you have to write up the solutions yourself, in your own words. Otherwise it is considered unauthorized aid or assistance (working too closely with another student on an individual assignment so that the end result is too similar), which is an academic offence under the University's Code of Behaviour on Academic Matters. If you find the solutions in books or on the internet, you must quote your source (and still write it up in your own words!) Otherwise, it may count as plagiarism which again is an academic offense.

Here is a sample latex source file and its pdf output for download, in case you want to type up your solutions in latex and need some hints.

If you like, you can attend Peer Study Team (PST) sessions for this course: Thursday 4-6, UC257. The sessions are being run by Phil Blakey and Mira Kuroyedov, who took this course last year.

The calendar' below will be updated as the course is underway. For example, I will give a few additional problems (not to be handed in), informations on tests and exams, required readings, and additional references.

 Calendar

Week 9: November 13-17, 2017
• Required reading (for this and next week): Section 3.2-3.4 (you can skip p.176-179)
• I have also posted some of my course notes on Blackboard (the notes that I'm lecturing off). They may contain mistakes, use at your own risk.
• Homework #7 is due 11pm on Friday, November 17
• Additional homework (do not hand in).
• Section 3.1, Problems 1
• Section 3.2, Problems 1,2b,2d,2f,4a,5,19,21,22
• Just for fun: Let $X$ be a set with $n\ge 3$ elements, and $S_1,\ldots,S_m$ proper subsets of $X$ (thus, $S_j\not=X$) such that every pair of distinct elements of $X$ is contained in exactly one of the $S_j$. Prove that $m\ge n$. (Hint: introduce a suitable incidence matrix, and consider ranks. This problem is quite challenging!)
Week 10: November 20-24, 2017
• Required reading: Section 4.1, 4.2 . We've spend one lecture on dual spaces'; you may want to review my notes on this (will be posted on Blackboard under course materials).

• Material covered this week: The general theory of linear systems of equations. Reduced row echelon form of a matrix. How to read off a basis of the null space and range of a matrix from its reduced row echelon form. How to determine the solution set of $Ax=b$ from the reduced row echelon form of $(A|b)$. Dual spaces, dual bases, dual maps and their matrices.
• Homework #8 is due 11pm on Monday, November 27
• Additional homework (do not hand in).
• Section 3.3, Problems 1, 2, 4, 10
• Section 3.4, Problems 1, 2 (pick a few)
• Section 4.1, problems 1, 3, 8, 9,
• (1) Prove that if $A,B\in M_{m\times n}(F)$ are in reduced row echelon form, and with $N(L_A)=N(L_B)$, then $A=B$.
• (2) Prove that the reduced row echelon form of a matrix is uniquely determined by the matrix, and does not depend on the order of the row operations used. Hint: Use (1) above.

Week 11: November 27 - December 1, 2017
• Required reading: We will be treating determinants a bit different from the textbook. An online reference along similar lines is Denton-Waldron: p.96-p.138; I will also post my notes.

• Material covered this week: Definition of determinant as a multi-linear form on column vectors, the formula as a sum over pemutations. Properties of determinants: Row and column operations, $\det(A^t)=\det(A)$ , $\det(AB)=\det(A)\det(B)$. The fact that $\det(A)\neq 0$ if and only if $A$ invertible. Computing $\det(A)$ be reducing to upper triangular form.
• Homework #9 is due 11pm on Monday, December 4
• Additional homework (do not hand in).
• Section 4.2, problems 1, 2, 5, 7, 9, 11. Also compute the last few determinants using the method explained on p. 234 (using that the determinant of an upper/lower triangular matrix is the product of diagonal entries).
• Section 4.3, problems 22. (We've encountered the Vandermonde matrix as the change-of-basis matrix from the standard basis of polynomials to the basis given by Lagrange polynomials. Now it's time to calculate its determinant.)
• Section 4.3, problem 28, Section 4.4, problem 1, Section 4.5, problems 3-10

Week 12: December 4-6, 2017
The final exam is on December 18, 2-5 pm. The official source for time, location, etc is the Arts and Sciences webpage. Please note that there are three exam locations; you have to go to the correct room depending on the first letter of your last name. I've posted last year's exam on Blackboard, under course materials. This year's exam will be a somewhat similar format. For further practice, you can use exams from earlier years on Dror Bar-Natan's MAT240 websites.