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Areas to be covered include Vector spaces: subspaces, basis, dimension. Systems of linear equations. Linear transformations: kernel, image, rank, matrix representation. Change of basis. Determinants. Eigenvalues and eigenvectors, characteristic polynomial, Cayley-Hamilton theorem, diagonalization. Introduction to inner products, Cauchy-Schwarz inequality, Gram-Schmidt orthogonalization.
Exclusion: (MATA40, MATA45); MAT147, 223, 224, 225, 229, 240, 247
Prerequisite: Grade 13 Calculus and one other Grade
13 Mathematics or OAC Calculus and OAC Algebra
Areas to be covered include, Rn as a model space in economics, constraints, functions of one and several variables, graphs. Derivatives, partial derivatives, differentials, rules for differentiation in several variables including chain rule, higher derivatives. Exponential and logarithm, geometric series, discrete and continuous interest. One variable optimization: relative and absolute extreme, graph sketching, word problems. Matrix algebra, linear equations. Optimization in several variables. Contour maps. Lagrange multipliers. Linear programming. This course is illustrated throughout by examples drawn from Economics. Students must have an approved calculator with memory and the exponential and logarithmic functions.
Exclusion: MATA26Y (MATA27Y), MATA29Y, MATA24, MATA25, MATB41; MAT123, 124, 125, 126, 133, 135, 137
Prerequisite: OAC Calculus & [OAC Algebra and Geometry or OAC Finite Mathematics]
Corequisite: ECOA02Y
NOTE:
This course is intended for certain students in some Management
and/or Economics Programmes. It does not satisfy the requirements
for any Programme in the Physical Sciences, and it may not be
used as a prerequisite for any further Mathematics course. Students
should consult with the Faculty in Management and Economics or
with Academic Advising to make sure that MATA28Y is appropriate
to their needs.
Projective geometry: Projective spaces, homogeneous coordinates and their transformations, duality, cross ratio, perspectivities, the fundamental theorem, conies. Models of non-Euclidean geometries. Introduction to finite geometries and coding theory. Students are encouraged to take MATB30F and MATC30S in the same year if possible.
Exclusion: (MATB25Y)
Prerequisite: MATB30H
An elementary introduction to a modern and fast-developing area of mathematics, through lectures and computer laboratories. One-dimensional dynamics: iterations of quadratic polynomials. Dynamics of linear mappings, attractors. Bifurcation, Henon map, Mandelbrot and Julia sets. History and applications.
Exclusion: MATC01, MAT335
Prerequisite: MATB43
Metric spaces, completeness, uniform convergence. Topics in measure theory: the Lebesgue integral, Riemann-Stieltjes integral, Lp spaces, Fourier series.
Exclusion: MAT337 (338)
Prerequisite: MATB42H & MATB43H
Basic counting principles, generating functions, permutations with restrictions. Fundamentals of graph theory with algorithms; applications (including network flows). Combinatorial structures including block designs and finite geometries.
Exclusion: (MATC31), MAT344
Prerequisite: MATB24H
Attention will be given to: fields of quotients, residue fields, finite fields, algebraic and transcendental field extensions, primitive elements, Galois extensions, review of group theory, the fundamental theorem of Galois theory, calculations and examples, the classical problems of angle trisection and construction of regular polygons.
Exclusion: (MAT447)
Prerequisite: MATC02, (MATC91H, MATB32)
Sets and functions, Zorn's lemma, cardinal arithmetic, Schroder-Bernstein theorem. Lebesque measure and integration, convergence theorems, derivatives and integrals. Continuity compactness and connectedness in metric and topological spaces, Baire category. Banach spaces and the basic tools of functional analysis. Hilbert space and linear operators.
Exclusion: (MATC50H), (MATC55H), MAT337, (338), (350)
Prerequisite: MATB32H (MATB49H) &
MATB41H & MATB42H & MATB43H
One or more topics in analysis such as: integral equations and Fredholm operators; Hilbert spaces; the prime number theorem; the gamma function; Fourier analysis; cardinality, metric spaces and function spaces.
Exclusions: MATC53Y, MAT357Y
Prerequisite: MATB43H
Complex arithmetic. Polynomials and elementary functions. Differentiation and the Cauchy Riemann equations. Cauchy's integral formula for differentiable functions and their Taylor expansion. Properties of analytic functions including Liouville's theorem, identity theorem, maximum modulus theorem and open mapping theorem. Laurent expansion and classification of isolated singularities. Residue calculus.
Exclusion: MAT334, 357
Prerequisite: MATB42H
(Formerly MATB32)
An introduction to rings and fields, covering the standard topics of integral domains; ideals, quotient rings and homomorphisms; polynomial rings and factorization; divisibility in domains (unique factorization domains and euclidean domains); extension fields. These concepts are then applied to geometric constructions (i.e. impossibility of constructing angle trisectors by ruler and compass), finite fields and algebraic coding theory.
Exclusion: (MATB32), MAT301
Prerequisite: MATA23
Co-requisite: MATB24
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