School of Mathematics School of Mathematics
Module MA1212 - Linear algebra II 2010-11 ( JF Mathematics, JF Theoretical Physics )
Lecturer: Dr. Vladimir Dotsenko
Requirements/prerequisites: prerequisite: MA1111
Duration: Hilary term, 11 weeks

Number of lectures per week: 3 lectures including tutorials per week

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

ECTS credits: 5
End-of-year Examination: 2 hour end of year examination.


  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.


Homework assignments will be handed out in class every week. Besides just obtaining answers to questions, you are supposed to justify your answers (in particular, every ``yes/no'' question also assumes the ``why'' question). Homeworks are due to hand in after Tuesday's classes; on the same evening solutions shall be posted on the course webpage, so late assignments are not accepted.


An in-class midterm test plus an exam in the end of the year plus the continuous assessment.

The final mark is 100%*final exam mark or 70%*final exam mark + 15%*continous assessment mark + 15% of the midterm test mark, whichever is higher.

Web page

Homework assignments, selected solutions, various handouts and announcements will be posted on the course web page

Learning Outcomes: On successful completion of this module, students will be able to:

Jul 27, 2011

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On 27 Jul 2011, 09:39.