G12VSP: VECTOR SPACES

• Module Information
• Handouts
• Coursework
• Assessment
• Miscellaneous links of interest

Module information for 2000 to 2001

• Credits: 10
• Duration: approximately 21 lectures (including one revision session), two lectures a week plus problem classes in Spring Semester, starting Monday 29/1/2001
• Lecturers
  • Dr J. Zacharias, room C102, email Joachim.Zacharias@nottingham.ac.uk
  • Dr J. F. Feinstein, room C301, email Joel.Feinstein@nottingham.ac.uk
• Postgraduate assistants: (marking and problem classes) L. Orton, V. Paskunas, T. Poole and T. Womack.
• Lecture times: Monday 3, Thursday 2 in M&P Building room B1.
• Problem classes: problem classes will take place fortnightly on Tuesdays in room C4 at 11:00 and 12:00. The dates of the problem classes will be 13/2, 27/2, 13/3, 24/4 and 8/5. The arrangements will be the same as for G12CAN: if you are taking HGBFLU Fluid Mechanics then you must attend the 11:00 problem class. If you are not taking HGBFLU then you should attend as follows: 11:00 surnames N-Z; 12:00 surnames A-M. For these problem classes, please sit only in rows 1, 2, 4, 5, 7, 8 or 10 counting from the front, so that the people taking the problem class can get to you. A register will be taken at each problem class. If you have a genuine reason for not being able to attend at the time you are assigned, you may attend at the other time: let us know in writing your name, the time that you wish to attend, and the reason that you can not attend at the given time.
• Provisional office hours (Spring Semester)
  • Dr J. Zacharias, room C102: Dr Zacharias is in his office at most times during the working day, and is available to answer questions unless he is busy. Specifically, he expects to be available on Mondays 16:00-18:00 and Fridays 14:00-16:00.
  • Dr J.F. Feinstein, room C301: Dr Feinstein expects to be in his office and available on Wednesdays and Thursdays from 9.30 AM to 12.30 PM. (See the web page http://www.maths.nott.ac.uk/personal/jff/ttnow.html for more details.)
• Brief content description: In this module we study abstract vector spaces and the maps between them. This material is of great importance in pure mathematics, and has widespread applications throughout mathematics. We study the connections between linear maps, bases and matrices in detail. We also generalize our notions of length, right-angles and projections by looking at inner product spaces. Some familiar examples and results from earlier modules will be examined from a more abstract point of view as part of the more powerful, general theory. There will be more emphasis on rigorous proof than in some earlier modules. Some results that you have met before, but which were not proved in earlier modules, will be fully justified here.

• Prerequisites: G1AMAL or some equivalent introductory course on linear algebra.
• Module aims: To teach the theory of abstract vector spaces and linear maps between them, building on the concrete examples and methods met in earlier modules, and providing a solid basis for later modules.
• Module objectives: That the student should:
  • be able to apply the definitions and theorems presented in the module to the solution of simple unseen problems;
  • be able to state and prove the principal theorems proved in the module;
  • understand the notions of vector space and basis and be able to calculate the dimension of elementary examples;
  • understand the notion of linear map and be able to find the matrix representation of a linear map with respect to given ordered bases;
  • be able to check a given square matrix for diagonalizability, and (where possible) to diagonalize it;
  • be able to apply the Gram-Schmidt process to find an orthonormal basis of a finite-dimensional subspace of an inner product space;
  • have a knowledge of a good range of concrete examples.
  
  Brief Syllabus:
  • Chapter 1. Vector spaces and subspaces Definitions and examples. Standard facts and their deductions from the vector space axioms. Intersections and sums of subspaces.
  • Chapter 2. Linear independence, spans and bases Linear spans, linear dependence and linear independence. Spanning sets. Finite-dimensional vector spaces and infinite-dimensional vector spaces. Bases for finite-dimensional vector spaces.
  • Chapter 3. Linear Maps Linear maps (also called linear transformations or linear homomorphisms). The kernel (or null space) and image of a linear map. Restrictions of linear maps to subspaces. Definitions of rank and nullity. The rank-nullity theorem. Composition of linear maps. Isomorphisms (linear) and isomorphic vector spaces.
  • Chapter 4. Matrices representing linear maps Ordered bases and vectors of coordinates. The matrix representing a linear map. Extension by linearity: how to define a linear map given its values at a basis. Transition matrices for changing basis.
  • Chapter 5. Diagonalization Similar matrices. Endomorphisms. The characteristic polynomial, characteristic equation, eigenvalues, and trace of a square matrix. Eigenvalues and eigenvectors for endomorphisms. Diagonalizable square matrices. The method for diagonalizing a matrix (in terms of eigenvectors). How to calculate powers of a diagonalizable matrix A.
  • Chapter 6. Inner product spaces Inner products and inner product spaces. Orthogonality of vectors. The Hermitian adjoint of a square matrix. Hermitian matrices (self-adjoint) and some of their elementary properties. Orthogonal bases and orthonormal bases for finite-dimensional subspaces of an inner product space. The Gram-Schmidt process.
  • Chapter 7. Further topics These may include, for example; diagonalization of quadratic forms and curve sketching; diagonalization of Hermitian matrices; minimal polynomials and the Cayley-Hamilton theorem.
    
  
  
• Books: The following books are all worth looking at. The most appropriate book for this module is the book by Bowers.
  
  1. R B Allenby, Linear Algebra, Modular Mathematics, Arnold paperback.
  2. T S Blyth and E F Robertson, Matrices and vector spaces,
  3. Essential Student Algebra, Volume 2, Chapman and Hall, 1986.
  4. T S Blyth and E F Robertson, Linear Algebra, Essential Student Algebra, Volume 4, Chapman and Hall, 1986.
  5. J Bowers, Matrices and Quadratic Forms, Modular Mathematics, Arnold paperback
  6. C W Curtis, Linear Algebra, Springer-Verlag.
  7. P R Halmos, Finite-dimensional vector spaces, Springer-Verlag.
  8. I N Herstein and D J Winter, A primer on Linear Algebra, Macmillan 1988.
  9. S. Lang, Introduction to Linear Algebra, 2nd edition, Springer-Verlag 1989.
  10. S. Lang, Linear Algebra, 3rd edition, Springer-Verlag 1987.
  11. S MacLane and G Birkoff, Algebra, 2nd edition (more advanced and abstract), American Mathematical Society 1999.
  12. Larry E. Smith, Linear Algebra, Springer-Verlag.

Handouts

Most handouts will be issued in printed form, and will also be available (as they are issued) from the module web page.

• Module information sheet  (this document).
• 1997-8 blow by blow for the module, as Lectured by Dr Feinstein (in 1998-9 the module was given by a visiting academic) ps, pdf
• 1999-2000 blow by blow for the module ps, pdf
• 2000-2001 blow by blow for the module now complete: ps, pdf
• Chapter 1, Vector Spaces and Subspaces: ps, pdf
• Chapter 2, Linear independence, spans and bases: ps, pdf
• Proof of the Exchange Lemma (Theorem 2.10): ps, pdf
• Chapter 3, Linear maps:
  • Part I: ps, pdf
  • Part II: ps, pdf
• Chapter 4, Matrices representing linear maps:
  • Part I, Ordered bases and coordinate systems: ps, pdf
  • Part II, The matrix representing a linear map: ps, pdf
  • Part III, Transition matrices and the change of basis formula: ps, pdf
• Chapter 5, Diagonalization of matrices:
  • Parts I-II, Square matrices; Similar matrices: ps, pdf
  • Parts III-IV, Eigenvalues and eigenvectors; Diagonalization: ps, pdf
• Chapter 6, Inner product spaces: ps, pdf

Course work

• Exercises for the enthusiast, with prizes! Please note that we did not personally invent these questions. (They are well-known, and mostly rather hard): ps, pdf
  Prize question 5 was correctly solved by David Roberts
  

Assessment

Miscellaneous links which may be of interest:

• G13MTS: Metric and topological spaces [an error occurred while processing this directive]

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