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# Department of Mathematical Sciences

# MATH1071 Linear Algebra I

Techniques from linear algebra are used in all of mathematics. This course gives an introduction to all the major ideas in the topic. The things you learn in this course will be very useful for most modules you take later on.

The first term is concerned with the solution of linear equations and the various ways in which the ideas involved can be interpreted including those given by matrix algebra, vector algebra and geometry. This enables us to determine when a system of equations has a unique solution and gives us a systematic way of finding it. These ideas are then developed further in terms of the theory of vector spaces and linear transformations. We will discuss examples of linear transformations that are familiar from geometry and calculus.

Any linear map can be put into a particularly easy form by changing the basis of the space on which it acts. The second term begins with the solution of the eigenvalue problem which tells you how to find this basis. We then go on to generalise the notions of length, distance and angle to any vector space. These ideas may be used in a surprisingly large range of contexts. We show how all these ideas come together in the applications to geometry and calculus introduced in the first term.

Throughout the course we will also discuss examples for the notion of a group, which is one of the fundamental organizing objects in mathematics.

## Outline of Course

Aim: To provide an introduction to all the major ideas in linear algebra.

### Term 1

• Vectors in ℝn (6 lectures): Vectors, addition and scalar multiplication in ℝn with concrete examples in ℝ² and ℝ³. Scalar product, vector product, triple product. Equations of lines and planes, linear systems of equations in 3 variables. Examples: scalar and vector equations of lines and planes in ℝ³.
• Linear Systems and Matrices (5 lectures): Arbitrary linear systems of equations, Gauss-Jordan elimination. Solutions of linear equations as generalisations of lines and planes in ℝ³. Multiplication and inversion of matrices. Gauss-Jordan elimination using matrix notation.
• Determinants and Groups I (6 lectures): Determinants and explicit methods for their calculation(row and column expansion). Properties of determinants. Axioms of groups. Examples: symmetric groups, GL(n), SL(n). The determinant in terms of permutations. Examples: areas of parallelograms, volumes of parallelepipeds.
• Vector spaces (7 lectures): Vector spaces and subspaces over ℝ. Examples: lines and planes in ℝ³. Linear independence, spanning sets, bases and coordinates, dimension. Vector spaces of polynomials. Affine subspaces. ℂn as a vector space.
• Linear mappings (6 lectures): Definition of linear mapping, matrices as linear mappings in ℝn (examples: dilations, projections, reflections, rotations in ℝ² and ℝ³). Differentiation and integration as a linear mapping (example: polynomials). Representation of linear mappings by matrices. Composition of linear mappings and matrix multiplication. Kernel (row and column), rank and image of a linear mapping.

### Term 2

• Change of basis and diagonalisation (7 lectures): Change of basis and of coordinates for linear maps. Eigenvalues and eigenvectors. Explicit calculation with characteristic polynomial. Diagonalisation by change of basis.
• Inner product spaces (8 lectures): Definition and examples: ℝn, ℂn, polynomials. Cauchy-Schwarz inequality. Orthonormal bases and Gram-Schmidt procedure. Orthogonal and unitary matrices. Examples: projection, reflections and distances in ℝ² and ℝ³. Orthogonal complement of a subspace. Diagonalisation of symmetric matrices by orthogonal matrices.
• Linear differential operators (3 lectures): 2nd order linear differential operators. Special polynomials as eigenfunctions.
• Groups II (4 lectures): More examples of linear groups: O(n), U(n). Modular arithmetic. Matrix realisation of symmetry groups of polygons (ℤn, dihedral groups).

### Prerequisites

For details of prerequisites, corequisites, excluded combinations, teaching methods, and assessment details, please see the Faculty Handbook.