Unit information: Linear Algebra and Geometry in 2013/14

Unit name	Linear Algebra and Geometry
Unit code	MATH11005
Credit points	20
Level of study	C/4
Teaching block(s)	Teaching Block 4 (weeks 1-24)
Unit director	Dr. Schubert
Open unit status	Not open
Pre-requisites	None (a good pass in A level Mathematics or equivalent is required)
Co-requisites	None
School/department	School of Mathematics
Faculty	Faculty of Science

Description including Unit Aims

Linear Algebra and Geometry begins with the straightforward ideas of real and complex numbers and their algebraic properties. Further, it introduces vectors and matrices, and develops the abstract notion of vector spaces as well as studies basic geometric objects in vector space, such as lines, hyperplanes, some standard curves and surfaces. This is one of the basic structures of pure mathematics; yet the methods of the course are also fundamental for applied mathematics and statistics.

Aims:

Mathematics 11005 aims to provide some basic tools and concepts for mathematics at the undergraduate level, with particular emphasis on fostering students' ability to think clearly and to appreciate the difference between a mathematically correct treatment and one that is merely heuristic; introducing rigorous mathematical treatments of some fundamental topics in mathematics.

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.

Relation to Other Units

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

Intended Learning Outcomes

At the end of the unit,the students should:

• be able to distinguish correct from incorrect and sloppy mathematical reasoning;

• 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.

Transferable Skills:

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

Teaching Information

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

Assessment Information

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%.

Reading and References

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.