### M1: Linear Algebra I (2022-23)

Course Term: Michaelmas
Course Lecture Information: 14 lectures
Course Overview:
Linear algebra pervades and is fundamental to algebra, geometry, analysis, applied mathematics, statistics, and indeed most of mathematics. This course lays the foundations, concentrating mainly on vector spaces and matrices over the real and complex number systems. The course begins with examples in $$\mathbb{R}^2$$ and $$\mathbb{R}^3$$, and gradually becomes more abstract. The course also introduces the idea of an inner product, with which angle and distance can be introduced into a vector space.
Learning Outcomes:
By the end of the course, students will be able to:
(i) use the definitions of a vector space, a subspace, linear dependence and independence, spanning sets and bases, both within the familiar setting of $$\mathbb{R}^2$$ and $$\mathbb{R}^3$$ and also for abstract vector spaces, and prove results using these definitions;
(ii) use matrices to solve systems of linear equations and to determine the number of solutions of such a system;
(iii) solve a range of problems relating to linear maps between vector spaces, thinking of linear maps abstractly or representing them using matrices as appropriate.
Course Synopsis:
Systems of linear equations.
Matrices and the beginnings of matrix algebra.
Use of matrices to describe systems of linear equations.
Elementary Row Operations (EROs) on matrices.
Reduction of matrices to echelon form.
Application to the solution of systems of linear equations.

Inverse of a square matrix.
Reduced row echelon (RRE) form and the use of EROs to compute inverses; computational efficiency of the method.
Transpose of a matrix;
orthogonal matrices.

Vector spaces: definition of a vector space over a field (such as
$$\mathbb{R}$$, $$\mathbb{Q}$$, $$\mathbb{C}$$).
Subspaces.
Many explicit examples of vector spaces and subspaces.

Span of a set of vectors.
Examples such as row space and column space of a matrix.
Linear dependence and independence.
Bases of vector spaces; examples.
The Steinitz Exchange Lemma; dimension.
Application to matrices: row space and column space, row rank and column rank.
Coordinates associated with a basis of a vector space.

Use of EROs to find bases of subspaces.
Sums and intersections of subspaces; the dimension formula.
Direct sums of subspaces.

Linear transformations: definition and examples (including projections associated with direct-sum decompositions).
Some algebra of linear transformations; inverses.
Kernel and image, Rank-Nullity Theorem.
Applications including algebraic characterisation of projections (as idempotent linear transformations).

Matrix of a linear transformation with respect to bases.
Change of Bases Theorem.
Applications including proof that row rank and column rank of a matrix are equal.

Bilinear forms; real inner product spaces; examples.
Mention of complex inner product spaces.
Cauchy--Schwarz inequality.
Distance and angle.
The importance of orthogonal matrices.