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Linear Algebra and Geometry (MATH 11005)

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Contents of this document:

Administrative Information

  1. Unit number and title: MATH 11005 Linear Algebra and Geometry
  2. Level: C/4 (Honours)
  3. Credit point value: 20 credit points
  4. Year: 12/13
  5. First Given in this form: 07/08 (similar material has been given for many years)
  6. Unit Organiser: Roman Schubert
  7. Lecturer: Dr. R. Schubert
  8. Teaching block: 1 and 2
  9. Prerequisites: An A in Mathematics A-level or equivalent.

Unit aims

Mathematics 11005 aims to provide some basic tools and concepts for mathematics at the undergraduate level, to develop clear mathematical thinking and to introduce rigorous mathematical treatments of some fundamental topics in mathematics.

General Description of the Unit

Mathematics 11005 begins with the complex plane, conics, and hyperplanes in n-space, which leads to the straightforward ideas of vectors and matrices, and develops the abstract notion of vector spaces. This is one of the basic structures of pure mathematics; yet the methods of the course are also fundamental for applied mathematics and statistics.

Relation to Other Units

Mathematics 11005 provides foundations for all other units in the Mathematics Honours programmes.

Teaching Methods

Lectures supported by lecture notes, problem sheets and small-group tutorials.

Learning Objectives

At the end of the unit, the students should:

  • have developed some familiarity with abstract mathematical thinking;
  • be familiar with geometric objects like lines, planes and hyperplanes, and their axiomatic generalisation into vector spaces and linear maps;
  • be able to solve linear equations using elementary operations;
  • be able to work with matrix algebra, including matrix inverses, determinants, and eigenvalues and eigenvectors.

Assessment Methods

The final assessment mark for the unit is constructed from two unseen written examinations: a January mid-sessional examination (counting 10%) and a May/June examination (counting 90%). Calculators and notes are NOT permitted in these examinations.

  • The mid-sessional examination in January lasts one hour. There are two parts, A and B. Part A consists of 4 shorter questions, ALL of which will be used for assessment. Part B consists of three longer questions, of which the best TWO will be used for assessment. Part A contributes 40% of the overall mark for the paper and Part B contributes 60%.
  • The summer examination in May/June lasts two-and-a-half hours. There are again two parts, A and B. Part A consists of 10 shorter questions, ALL of which will be used for assessment. Part B consists of five longer questions, of which the best FOUR will be used for assessment. Part A contributes 40% of the overall mark for the paper and Part B contributes 60%.

Award of Credit Points

To be awarded the credit points for this unit you must normally pass the unit, i.e. you must achieve an assessment mark of at least 40.

The assessment mark is calculated as described in the Assessment section above. Details of the university's common criteria for the award of credit points are set out in the Regulations and Code of Practice for Taught Programmes at http://www.bristol.ac.uk/esu/assessment/codeonline.html

Note that for this unit:

  • first year students are expected to attend all the relevant tutorials,
  • all students are expected to hand in attempts to the weekly exercises set.

Transferable Skills

Clear logical thinking; clear mathematical writing; problem solving; the assimilation of abstract and novel ideas.

Texts

There are many good linear algebra texts. They come in different styles, some follow a more abstract approach, others emphasise applications and computational aspects. Some students may prefer the style of one book more than another.

The following is a selection of textbooks which cover a variety of styles:

  • G. Strang, "Linear Algebra and its Applications".
  • R. Allenby, "Linear Algebra"
  • H. Anton and C. Rorres, "Elementary Linear Algebra"
  • S. Lang, "Linear Algebra"
  • S. Lipschutz and M. Lipson, "Linear Algebra"
The lectures will present the material in a different order from most textbooks.

Syllabus

Note: topics may not appear in exactly this order.

  1. Cosine, sine, and the complex numbers.
  2. Conic sections.
  3. Distance, lines, and hyperplanes in n-space.
  4. The solution of linear equations using the three elementary operations.
  5. Linear transformations from n-space to m-space; surjectivity, injectivity, and kernels.
  6. Matrices and matrix algebra; representing linear transformations from n-space to m-space using matrices; solving matrix equations using elementary matrices; inverses using elementary matrices.
  7. Determinants; connections with elementary matrices.
  8. Vector spaces and their basic properties.
  9. Subspaces of vector spaces; linear combinations and span.
  10. Linear dependence and independence; application to rows (columns) of matrices.
  11. Bases for a vector space; dimension of a vector space; row and column rank of a matrix; equality of row and column rank.
  12. Linear transformations from one vector space to another; using matrices to represent linear transformations from one finite-dimensional vector space to another.
  13. Rank and nullity of a linear transformation, and the relationship between them.
  14. Eigenvalues and eigenvectors; characteristic polynomial of a matrix.
  15. Diagonalisation of a matrix; properties of real symmetric matrices.
  16. Inner products and inner product spaces; symmetric and orthogonal matrices; diagonalisation of matrices in inner product spaces.