MA 1111/1212: Linear Algebra

2009-2010

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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).

Exam materials: sample papers, actual papers, solutions

Sample midterm paper [PS] [PDF]
Midterm paper [PS] [PDF]
Solutions to the midterm paper [PS] [PDF]
Sample final paper [PS] [PDF]
You may also check papers for two previous years 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!).

NEW! 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!)
NEW! The most relevant reading on vectors and quaternions would be the following extract from lecture notes by Dr. David Wilkins for MA2C02 course: [PDF]

Handouts

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

Orientation notes on the course: syllabus, assessment etc. (more or less a copy of this webpage) [PS] [PDF]
Linear operators on a finite-dimensional vector space: a brief HOWTO [PS] [PDF]
Intersections and relative bases [PS] [PDF]
Jordan normal form theorem [PS] [PDF]
Examples on Jordan forms for nilpotent operators [PS] [PDF]
Examples on computations with Jordan normal forms [PS] [PDF]
Orthonormal bases, orthogonal complements, and orthgonal direct sums [PS] [PDF]
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]
Solution to Q5 from the final exam (author: Jack Kelly) [PS] [PDF]

Assessment etc.

For TSM students, your final grade is either the grade on the final exam or 80% of final exam plus 20% of home assignments grade, whichever is larger. For non-TSM students, your final grade is either the grade on the final exam or 70% of final exam plus 15% of home assignments grade plus 15% of the midterm test result, whichever is larger. In short, if we denote by H your grade on homework, by MT your midterm result, and by E your final exam result, your overall grade will be obtained from the formula max(0.2H+0.8E,E) for TSM students, and max(0.15H+0.15MT+0.7E,E) for non-TSM students.