Linear algebra, TCD 2010/11
Syllabus
MA1111- Linear algebra in 2d and 3d. Vectors. Dot and cross products.
- Systems of simultaneous linear equations. Gauss--Jordan elimination.
- (Reduced) row echelon form for a rectangular matrix. Principal and free variables. Matrix product and row operations. Computing the inverse matrix using row operations.
- Permutations. Odd and even permutations. Determinants. Row and column operations on determinants. det(AT)=det(A).
- Minors. Cofactors. Adjoint matrix. Computing the inverse matrix using determinants. Cramer's rule for systems with the same number of equations and unknowns.
- Fredholm's alternative. An application: the discrete Dirichlet's problem.
- Coordinate vector space. Linear independence and completeness.
- Fields: rationals, reals, and complex. Abstract vector spaces. Linear independence and completeness in abstract vector spaces. Bases and dimensions. Subspaces.
- 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.
- Kernels and images. Ranks. Dimension formulas.
- Characteristic polynomials. Eigenvalues and eigenvectors. Diagonalisation in the case when all eigenvalues are distinct.
- Cayley--Hamilton theorem. Minimal polynomial of a linear operator. Examples (operators with A2=A).
- Invariant subspaces. An application: two commuting linear operators have a common eigenvector. Direct sums.
- Normal form of a nilpotent operator. Jordan normal form (Jordan Decomposition Theorem). Applications: closed expressions for Fibonacci numbers and other recursively defined sequences.
- Orthonormal bases; Gram--Schmidt orthogonalisation. Orthogonal complements and orthogonal direct sums. Bessel's inequality.
- Bilinear and quadratic forms. Sylvester's criterion. The law of inertia. Spectral Theorem for symmetric operators.
Exam materials: sample papers, actual papers, solutions
Sample midterm paper [PDF]Solutions to the sample midterm paper [PDF]
Midterm paper [PDF]
Solutions to the midterm paper [PDF]
Sample final paper for the 1111 part [PDF]
Sample final paper for the 1212 part [PDF]
Final paper for the 1111 part [PDF]
Final paper for the 1212 part [PDF]
Handouts
These handouts will be used during the year; they will be distributed in class, so you do not need to download them.
MA1111Vectors and quaternions: an extract from lecture notes by Dr. David Wilkins for MA2C02 course: [PDF]
Linearity on the example of dot and cross products [PDF]
Linear operators, matrices, change of coordinates: a brief HOWTO [PDF]
MA1212
Intersections and relative bases [PDF]
Examples on Jordan forms for nilpotent operators [PDF]
Jordan normal form theorem [PDF]
Examples on computations with Jordan normal forms [PDF]
Orthonormal bases, orthogonal complements, and orthgonal direct sums [PDF]
MA1111+1212
Some standard types of linear algebra questions [PDF]
Several problems in Linear Algebra (bonus questions for those who feel confident with the course) [PDF]
Textbooks
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's "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.
There are also some good online sources to check out, e.g. 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!) and 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!)
Assessment
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.