Systems of linear equations, matrix algebra, determinants. Vector geometry in R2 and R3. Complex numbers. Rn: subspaces, linear independence, bases, dimension, column spaces, null spaces, rank and dimension formula. Orthogonality, orthonormal sets, Gram-Schmidt orthogonalization process, least square approximation. Linear transformations from Rn to Rm. The determinant, classical adjoint, Cramer's rule. Eigenvalues, eigenvectors, eigenspaces, diagonalization. Function spaces and applications to a system of linear differential equations. The real and complex number fields.

Abstract vector spaces: subspaces, dimension theory. Linear mappings: kernel, image, dimension theorem, isomorphisms, matrix of a linear transformation. Change of basis, invariant subspaces, direct sums, cyclic subspaces, Cayley-Hamilton theorem. Inner product spaces, orthogonal transformations, orthogonal diagonalization, quadratic forms, positive definite matrices. Complex operators: Hermitian, unitary and normal. Spectral Theorem. Isometries of R2 and R3.

Differential and integral calculus of several variables: partial differentiation, chain rule, extremal problems, Lagrange multipliers, classification of critical points. Multiple integrals, Green's theorem and related topics.

"Bridging Course"; accepted as prerequisite for upper level courses in replacement of MAT232H5. Limited Enrolment. Sequences and series, power series, Taylor series, trigonometric and inverse trigonometric functions and their use in integrations. Differential and integral calculus of several variables; partial differentiation, chain rule, extremal problems, Lagrange multipliers, classification of critical points. Multiple integrals, Green's theorem and related topics.

The implicit function theorem, vector fields. Transformations. Parametrized integrals. Line, surface and volume integrals. Theorems of Gauss and Stokes with applications.

A theoretical approach to Linear Algebra and its foundations, aimed at students with a serious interest in Mathematics. Topics to be covered: Vector spaces over arbitrary fields (including C and finite fields), linear equations and matrices, bases and linear independence, linear transformations, determinants, eigenvalues and eigenvectors, similarity, change of basis, diagonalization, the characteristic and minimal polynomials, the Cayley-Hamilton theorem.

Ordinary differential equations of the first and second order, existence and uniqueness; solutions by series and integrals; linear systems of first order; linearization of non-linear systems. Applications in life and physical sciences. Power series solutions, boundary value problems, Fourier series solutions, numerical methods.

Continuation of MAT240H5. A theoretical approach to real and complex inner product spaces, isometries, orthogonal and unitary matrices and transformations. The adjoint. Hermitian and symmetric transformations. Spectral theorem for symmetric and normal transformations. Polar representation theorem. Primary decomposition theorem. Rational and Jordan canonical forms. Additional topics including dual spaces, quotient spaces, bilinear forms, quadratic surfaces, multilinear algebra.

A rigorous and proof-intensive course in multivariable calculus for students with a serious interest in mathematics.  Topology of metric spaces; compactness, functions and continuity, the extreme value theorem. Derivatives; inverse and implicit function theorems, maxima and minima. Integration; Fubini's theorem, partitions of unity, change of variables. Integration on manifolds; Stokes' theorem.  

Most applications of Mathematics involve the use of a computer. Numerical analysis studies how formulas can be transformed into computations. The topics covered may include: numerical methods in Calculus, such as series expansions and rates of convergence, numerical integration and differentiation, finite interpolation methods, splines; and numerical methods for ordinary differential equations, such as root-finding methods, Fourier series and Fourier transform, least-squares approximation, regression, and principal component analysis.

This course provides a richly rewarding opportunity for students in their second year to work in the research project of a professor in return for 299H course credit. Students enrolled have an opportunity to become involved in original research, learn research methods and share in the excitement and discovery of acquiring new knowledge. Participating faculty members post their project descriptions for the following summer and fall/winter sessions in early February and students are invited to apply in early March. See Research Opportunity Program (ROP) for more details.

This courses provides a richly rewarding opportunity for students in their second year to work in the research project of a professor in return for 299Y course credit. Students enrolled have an opportunity to become involved in original research, learn research methods and share in the excitement and discovery of acquiring new knowledge. Participating faculty members post their project descriptions for the following summer and fall/winter sessions in early February and students are invited to apply in early March. See Experiential and International Opportunities for more details.

Permutations and permutation groups. Linear groups. Abstract groups, homomorphisms, subgroups. Symmetry groups of regular polygons and platonic solids, wallpaper groups. Group actions, class formula. Cosets, Lagrange's theorem. Normal subgroups, quotient groups. Classification of finitely generated Abelian Groups. Emphasis on examples and calculations.

(Cross list with CSC322H5) The course will take students on a journey through the methods of algebra and number theory in cryptography, from Euclid to Zero Knowledge Proofs. Topics include: block ciphers and the Advanced Encryption Standard (AES); algebraic and number-theoretic techniques and algorithms in cryptography, including methods for primality testing and factoring large numbers; encryption and digital signature systems based on RSA, factoring, elliptic curves and integer lattices; and zero-knowledge proofs.

This course is an introduction to the theory of matrix groups with a particular emphasis on applications and examples. This course will cover orthogonal transformations in two and three dimensions, quaternions, isometries of Euclidean space, Lie algebras and matrix exponentials.

This course is an introduction to the theory of curves and surfaces with a particular emphasis on applications and computational techniques. This course will cover curves in R2 and R3, curvature, torsion, differential of maps, First Fundamental Form, Parallel transport, Bishop Frames, Geodesics, Gauss-Bonnet Theorem, and Gaussian curvature.

The relationships among axioms, proofs, consistency and truth in mathematics. Soundness and Completeness. Introductions to model theory, set theory, and computability; arithmetic as a central example. Gödel's incompleteness theorems; outlines of their proofs. This course emphasizes rigour.

Partial differential equations of applied mathematics, mathematical models of physical phenomena, basic methodology.

Elementary topics in number theory such as: prime numbers; arithmetic with residues; Gaussian integers, quadratic reciprocity law, representation of numbers as sums of squares. (This course emphasizes rigour).

The course will serve as an introduction to mathematical modelling of biological processes. It will cover a selection of the following topics: Difference equations and applications. Linear differential equations and systems; phase plane analysis; nonlinear systems of differential equations and linearization; Poincaré-Bendixson Theorem. Applications of differential equations to biology, including a logistic population with harvesting; predator-prey model; competing species; epidemic models. Examples of partial differential equations; reaction-diffusion equation; pattern formation.

Stability in nonlinear systems of differential equations, bifurcation theory, chaos, strange attractors, iteration of nonlinear mappings and fractals. This course will be geared towards students with interest in sciences.

Theory of functions of one complex variable: analytic and meromorphic functions; Cauchy's theorem, residue calculus. Topics from: conformal mappings, analytic continuation, harmonic functions.

The real numbers; Sequences and series; Functional limits; Topology in R^n; Differentiation and Integration; Power Series; Metric Spaces; Integrability and sets of measure zero. The course emphasizes rigour and theory.

Basic counting principles, generating functions, permutations with restrictions. Fundamentals of graph theory with algorithms; applications (including network flows).

Complex numbers, the complex plane and Riemann sphere, Möbius transformations, elementary functions and their mapping properties, conformal mapping, holomorphic functions, Cauchy's theorem and integral formula. Taylor and Laurent series, maximum modulus principle, Schwarz' lemma, residue theorem and residue calculus.

Introduction to a topic of current interest in applied mathematics. Content will vary from year to year. The contact hours for this course may vary in terms of contact type (L, T) from year to year, but will be between 36-48 contact hours in total. See the UTM Timetable.

Introduction to a topic of current interest in mathematics. Content will vary from year to year. The contact hours for this course may vary in terms of contact type (L, T) from year to year, but will be between 36-60 contact hours in total. See the UTM Timetable.

Introduction to a topic of current interest in mathematics. Content will vary from year to year. This course may include a tutorial and/or practical section in some years. The contact hours for this course may vary in terms of contact type (L, T) from year to year, but will be between 36-60 contact hours in total. See the UTM Timetable.

This is a one-term course to give students extensive practice in the writing of mathematics. The format will be to study excellent expositions of important ideas of mathematics and then to assign short writing assignments based on them.

Students explore a topic in mathematics under the supervision of a faculty member.