MA 1111/1212: Linear Algebra

2010-2011

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• Timetable
• Syllabus
  • MA1111
  • MA1212
• Materials
  • Homeworks
  • Tutorial papers
  • Exam materials: sample papers, actual papers, solutions
  • Selected answers and solutions for homeworks
  • Exam announcements
• Textbooks
• Handouts
• Assessment etc.
• Feedback

Important!There will be no class on Monday March 28.

Timetable

1111: we shall have classes on Mondays at 1pm and 5pm in Maxwell Lecture Theatre, Hamilton Building, and on Tuesdays at 5pm in Large Lecture Theatre in Chemistry Building.

1212: we shall have classes on Mondays at 1pm in Maxwell Lecture Theatre, Hamilton Building, Tuesdays at 5pm in Large Lecture Theatre in Chemistry Building, and Thursdays 12 noon in Joly Lecture Theatre (until March 17) and Lecture Theatre 11, Lloyd Building (from March 24).

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(A^T)=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

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

MA1212

Tutorial papers

MA1111

MA1212

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

MA1212

Exam announcements

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

Textbooks

Reading suggestions

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

Handouts

MA1111

MA1212

MA1111+1212

Assessment etc.

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.