Linear algebra, TCD 2008/09
Syllabus
- 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. 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. Ranks. Maximal size of nonzero minors is equal to the rank.
- 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.
- 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).
- An application: 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
A sample Michaelmas term paper [PDF]Solutions to the sample Michaelmas Term paper [PDF]
The Michaelmas Term paper [PDF]
Solutions to the Michaelmas Term paper [PDF]
A sample Hilary term paper [PDF]
Solutions to the sample Hilary Term paper [PDF]
The Hilary term paper [PDF]
Solutions to the Hilary Term paper [PDF]
A sample Trinity Term paper (warning: it is more time-consuming than the actual exam paper; I just tried to squeeze in enough things for you to practice!) [PDF]
Solutions to the sample Trinity Term paper [PDF]
The Trinity Term paper [PDF]
Solutions to the Trinity Term paper [PDF]
Handouts
These handouts will be used during the second and the third term; they will be distributed in class, so you do not need to download them.
Linear operators on a finite-dimensional vector space: a brief HOWTO [PDF]
Jordan normal form theorem [PDF]
Examples on computations with Jordan normal forms [PDF]
Orthonormal bases, orthogonal complements, and orthgonal direct sums [PDF]
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!).
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!)