MA 1111/1212: Linear Algebra

2010-2011

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Syllabus

MA1111

  1. Linear algebra in 2d and 3d. Vectors. Dot and cross products.
  2. Systems of simultaneous linear equations. Gauss--Jordan elimination.
  3. (Reduced) row echelon form for a rectangular matrix. Principal and free variables. Matrix product and row operations. Computing the inverse matrix using row operations.
  4. Permutations. Odd and even permutations. Determinants. Row and column operations on determinants. det(AT)=det(A).
  5. Minors. Cofactors. Adjoint matrix. Computing the inverse matrix using determinants. Cramer's rule for systems with the same number of equations and unknowns.
  6. Fredholm's alternative. An application: the discrete Dirichlet's problem.
  7. Coordinate vector space. Linear independence and completeness.
  8. Fields: rationals, reals, and complex. Abstract vector spaces. Linear independence and completeness in abstract vector spaces. Bases and dimensions. Subspaces.
  9. Linear operators. Matrix of a linear operator relative to given bases. Change of basis. Transition matrices. Similar matrices define the same linear operator in different bases. Example: a closed formula for Fibonacci numbers.

MA1212

  1. Kernels and images. Ranks. Dimension formulas.
  2. Characteristic polynomials. Eigenvalues and eigenvectors. Diagonalisation in the case when all eigenvalues are distinct.
  3. Cayley–Hamilton theorem. Minimal polynomial of a linear operator. Examples (operators with A2=A).
  4. Invariant subspaces. An application: two commuting linear operators have a common eigenvector. Direct sums.
  5. Normal form of a nilpotent operator. Jordan normal form (Jordan Decomposition Theorem). Applications: closed expressions for Fibonacci numbers and other recursively defined sequences.
  6. Orthonormal bases; Gram–Schmidt orthogonalisation. Orthogonal complements and orthogonal direct sums. Bessel's inequality.
  7. Bilinear and quadratic forms. Sylvester's criterion. The law of inertia. Spectral Theorem for symmetric operators.

Materials

Course materials on this webpage are in two different formats: PDF and PS (Postscript). PDF files are handled, for example, by Adobe Acrobat Reader. It makes sense to learn how to handle PS files as well. I suggest some public-domain software that opens these files: Ghostview, a user-friendly interface to Ghostscript (so you need both of these).

Homeworks

Important! Solutions to home assignments will be posted on this webpage on the day when they are due. As a consequence, I won't be able to accept late homeworks.

MA1111

Homework due October 5 [PS] [PDF]
Homework due October 12 [PS] [PDF]
Homework due October 19 [PS] [PDF]
Homework due October 26 [PS] [PDF]
Homework due November 2 [PS] [PDF]
Homework due November 16 [PS] [PDF]
Homework due November 23 [PS] [PDF]
Homework due November 30 [PS] [PDF]
Homework due December 7 [PS] [PDF]
Homework due December 13 [PS] [PDF]

MA1212

Homework due January 24 [PS] [PDF]
Homework due January 31 [PS] [PDF]
Homework due February 7 [PS] [PDF]
Homework due February 14 [PS] [PDF]
Homework due February 21 [PS] [PDF]
Homework due March 7 [PS] [PDF]
Homework due March 14 [PS] [PDF]
Homework due March 21 [PS] [PDF]
Homework due March 29 [PS] [PDF]

Tutorial papers

MA1111

Tutorial 1, October 5 [PS] [PDF]
Tutorial 2, October 19 [PS] [PDF]
Tutorial 3, November 2 [PS] [PDF]
Tutorial 4, November 23 [PS] [PDF]
Tutorial 5, December 7 [PS] [PDF]

MA1212

Tutorial 1, January 27 [PS] [PDF]
Tutorial 2, February 10 [PS] [PDF]
Tutorial 3, March 24 [PS] [PDF]

Selected answers and solutions to homeworks

Important! Solutions to home assignments will be posted on this webpage on the day when they are due. As a consequence, I won't be able to accept late homeworks.

MA1111

Solutions to the homework due October 5 [PS] [PDF]
Solutions to the homework due October 12 [PS] [PDF]
Solutions to the homework due October 19 [PS] [PDF]
Solutions to the homework due October 26 [PS] [PDF]
Solutions to the homework due November 2 [PS] [PDF]
Solutions to the homework due November 16 [PS] [PDF]
Solutions to the homework due November 23 [PS] [PDF]
Solutions to the homework due November 30 [PS] [PDF]
Solutions to the homework due December 7 [PS] [PDF]
Solutions to the homework due December 13 [PS] [PDF]

MA1212

Solutions to homework due January 24 [PS] [PDF]
Solutions to homework due January 31 [PS] [PDF]
Solutions to homework due February 7 [PS] [PDF]
Solutions to homework due February 14 [PS] [PDF]
Solutions to homework due February 21 [PS] [PDF]
Solutions to homework due March 7 [PS] [PDF]
Solutions to homework due March 14 [PS] [PDF]
Solutions to homework due March 21 [PS] [PDF]
Solutions to homework due March 28 [PS] [PDF]

Exam announcements

For the module 1212, there will be an in-class midterm test. It will take place on Tuesday March 8, from 17:15 till 18:45 in the Large Lecture Theatre in Chemistry Building (be there at 17:00 to be seated).

For each of the modules 1111 and 1212 there will be a two hour final exam in the end of the year.

Exam materials: sample papers, actual papers, solutions

Sample midterm paper [PS] [PDF]
Solutions to the sample midterm paper [PS] [PDF]
Midterm paper [PS] [PDF]
Solutions to the midterm paper [PS] [PDF]
Sample final paper for the 1111 part [PS] [PDF]
Sample final paper for the 1212 part [PS] [PDF]
Final paper for the 1111 part [PS] [PDF]
Final paper for the 1212 part [PS] [PDF]
You may also check papers for the previous years here, here, and here.

Textbooks

Reading suggestions

There will be no lecture notes for this course, so you are encouraged to take notes during the lectures — it takes effort but is really helpful. There are many books which you might find helpful, though they do not correspond exactly to the course content and the order of presentation of topics. For the first part of the course (Linear Algebra in 2d and 3d, systems of linear equations, operations with matrices), have a glance at Anton/Rorres' Elementary Linear Algebra (applications version). For the second part of the course (abstract vector spaces, linear operators, quadratic forms etc.) the exposition will be mostly close to the one from Gelfand's Lectures on Linear Algebra (there should be several copies in the College Library, also some 20 copies belonging to the School of Maths are in my office, and you may borrow them as well). You are also encouraged to attempt problems from Linear Algebra Problem Book by Paul Halmos (some of these problems are quite difficult!).

A copy of Gelfand's book turns out to be available on Google Books (here): some pages are missing, but most of it is there! Same is true for the Halmos's book: click here.

Online textbooks

Elementary Linear Algebra, lecture notes by Keith Matthews (this link is just for your information, you should not expect it to be much related to what happens in class!)
MIT Linear Algebra Course, you can find several useful essays on Linear Algebra there, as well as lots of problems with solutions. (Again, this course is different from what we have in class, so do not rely on these materials too much!)

Handouts

These handouts will be used during the year; they will be distributed in class, so you do not need to download them.

MA1111

Syllabus and organisational notes on the 1111 half of the course [PS] [PDF]
Vectors and quaternions: an extract from lecture notes by Dr. David Wilkins for MA2C02 course: [PDF]
Linearity on the example of dot and cross products [PS] [PDF]
Solutions to the 4th tutorial [PS] [PDF]
Linear operators, matrices, change of coordinates: a brief HOWTO [PS] [PDF]

MA1212

Syllabus and organisational notes on the 1212 half of the course [PS] [PDF]
Intersections and relative bases [PS] [PDF]
Examples on Jordan forms for nilpotent operators [PS] [PDF]
Jordan normal form theorem [PS] [PDF]
Examples on computations with Jordan normal forms [PS] [PDF]
Orthonormal bases, orthogonal complements, and orthgonal direct sums [PS] [PDF]

MA1111+1212

Some standard types of linear algebra questions [PS] [PDF]
Several problems in Linear Algebra (bonus questions for those who feel confident with the course) [PS] [PDF]

Assessment etc.

For the 1111 course, the final mark is 100%*final exam mark or 80%*final exam mark + 20%*home assignments result, whichever is higher.

For the 1212 course, the final mark is 100%*final exam mark or 70%*final exam mark + 15%*home assignments result + 15% of the midterm test result, whichever is higher.