MATH1540: University Mathematics for Financial Studies
Coordinate partial derivatives, total differentiability, the chain rule and local sensitivity models.
Partial derivatives and local models
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Chapter 1
Partial derivatives and local models
Distinguish coordinate rates from total linear approximations, apply the pointwise chain rule and check the domain and units of sensitivity models.
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Partial derivatives and local models
Distinguish coordinate rates from total linear approximations, apply the pointwise chain rule and check the domain and units of sensitivity models.
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Partial derivatives and gradients
Measure coordinate rates of change, organize them into the gradient, and understand higher partial derivatives and their hypotheses.
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Differentiability and linearization
Define a derivative through a linear approximation with a controlled remainder, and use continuous partial derivatives to establish differentiability.
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Chain rule and directional derivatives
Compose Jacobian matrices in the correct order and relate derivatives along curves to gradients, directional rates, and level sets.
Chapter 2
Linear systems
Linear systems
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Equations and solution sets
Treat a linear system as a set of simultaneous conditions, classify its solution set carefully, and see why equation operations preserve that set.
Chapter 3
Matrices and elimination
Matrices and elimination
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Matrix basics
Read matrices carefully as mathematical objects: size, entries, equality, basic operations, and how systems of equations become matrix statements.
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Augmented matrices and row operations
Translate a system into `Ax = b`, package it as `[A|b]`, and understand exactly why elementary row operations preserve the solution set.
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Gaussian elimination and RREF
Gaussian elimination preserves a system's solution set while exposing its pivot equations and free variables.
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Solution-set types
Use RREF to classify a linear system rigorously, identify pivot and free variables, and explain why a system has exactly one solution, infinitely many solutions, or no solution.
Chapter 4
Matrix algebra
Matrix algebra
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Matrix multiplication, identity matrices, and linear systems
Learn when matrix products are defined, how the row-by-column rule and identity matrices work, and how multiplication encodes linear systems.
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Transpose, symmetric, and skew-symmetric matrices
Use transpose, symmetry, and commuting products to read matrix structure and algebraic behavior.
Chapter 5
Solution structure
Solution structure
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Homogeneous systems and null space
The null space describes the differences between solutions, their free coordinates, and the structure of every consistent solution set.
Chapter 6
Invertibility and rank
Invertibility and rank
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RREF uniqueness and well-defined rank
Prove that every row-equivalence class has only one reduced row-echelon form, then use that theorem to make rank independent of the chosen reduction path.
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Invertible matrices
Invertibility connects reversible matrix actions, unique solutions for every right-hand side, and the algebra of inverse matrices.
Chapter 7
Spaces and bases
Spaces and bases
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Vector spaces
Define vector spaces carefully, compare the standard examples, and prove the first structural facts that every later span, basis, and dimension argument depends on.
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Subspaces
Use the subspace test carefully, prove the first structural consequences, and learn the standard matrix and equation-defined examples that recur throughout linear algebra.
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Linear combinations and span
Treat span as a precise set of outputs, connect it to linear systems, and learn how to decide whether a vector can or cannot be built from a given list.
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Linear dependence and independence
Detect redundancy in a list of vectors, test independence by direct relations and row reduction, and read the geometric meaning in low dimensions.
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Basis and dimension
A basis gives unique coordinates; dimension records the common size of all bases and provides criteria for spanning and independence.
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Column space, row space, and rank
Read a matrix as both a source of outputs and a source of linear relations, then use rank to count the genuinely independent directions that remain.
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Basis extension and change of basis
Turn the basis theorems into usable tools: extend independent lists, replace old basis vectors, and compare coordinates across two bases.
Chapter 8
Determinants
Determinants
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Determinants and cofactor expansion
Define determinants carefully through minors and cofactors, then learn how cofactor expansion turns one scalar into a precise summary of square-matrix structure.
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Row operations, products, and invertibility
Track exactly how row operations change determinants, then connect that behavior to multiplicativity, inverse matrices, and invertibility tests.
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Transpose, column operations, and Cramer's rule
Transpose transfers determinant identities from rows to columns; cofactor cancellation then gives the adjoint formula and Cramer’s rule.
Chapter 9
Eigenvalues and diagonalization
Eigenvalues and diagonalization
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Eigenvalues, eigenvectors, and eigenspaces
Define eigenvalues through the equation Av=λv, then recast the same idea as a null-space and determinant question so the structure becomes computable.
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Diagonalization and similarity
Treat diagonalization as a basis change built from eigenvectors, then use similarity to explain when a matrix can be simplified without changing its essential eigenvalue data.
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Characteristic polynomials and diagonalization tests
Use characteristic polynomials, algebraic and geometric multiplicity, and the distinct-eigenvalue test to decide when eigenvalue data is enough for diagonalization.
Chapter 10
Orthogonality
Orthogonality
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Inner products, norms, and angles
Define the standard inner product and norm on R^m, then connect those formulas to length, angle, and the first structural inequalities.
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Orthogonal sets and orthonormal bases
Use orthogonality to build orthogonal and orthonormal bases, then read coefficients without solving a linear system every time.
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Gram-Schmidt orthogonalization
Apply Gram-Schmidt to turn a basis into an orthogonal or orthonormal basis while preserving the same span.
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Cauchy-Schwarz and triangle inequalities
Study Cauchy-Schwarz and triangle inequalities as the two core estimates that control length, angle, and equality cases in inner-product spaces.
Chapter 11
Euclidean geometry
Euclidean geometry
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Vectors, inner products and distance
Develop Euclidean length, angle and orthogonal projection from coordinates, with proofs of the fundamental norm inequalities.
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Lines, planes and volumes
Translate affine geometry between parameters and equations, derive nearest-point formulas, and use cross and triple products for area and volume.
Chapter 12
Domains, limits and continuity
Domains, limits and continuity
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Domains and point sets
Describe multivariable domains using balls, interior, boundary, closure and paths, with proofs that connect the geometric language.
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Sequences and multivariable limits
Use convergence, the sequential criterion, curved paths and uniform estimates to establish or disprove limits in Euclidean space.
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Continuity and compactness
Study continuity relative to a domain, continuous extensions, compact sets, attained extrema and uniform continuity.
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Finite-dimensional compactness and attained extrema
Finite-dimensional compactness and attained extrema
Chapter 13
Taylor models and optimization
Taylor models and optimization
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Mean value estimates and Taylor approximation
Turn one-dimensional estimates along line segments into multivariable bounds and second-order approximations.
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Critical points and extrema
Find and classify stationary points, and distinguish local tests from global optimization.
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Constrained optimization and Lagrange multipliers
Find constrained candidates while checking regularity, existence and exceptional points.
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Taylor models: worked example and practice
Taylor models: worked example and practice
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All-order multivariable Taylor theorem
Multi-index coefficients, integral and Lagrange remainders, and the local Peano theorem, with full proofs and solved practice.
Chapter 14
Multiple integrals
Multiple integrals
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Double integrals and iterated integration
Build double integrals from area sums and choose an integration order from the geometry of a region.
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Triple integrals and three-dimensional coordinates
Set up volume integrals in Cartesian, cylindrical and spherical coordinates.
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Change of variables and Jacobians
Transform an integration region and its area scale without losing multiplicity or orientation.
Chapter 15
Integral proof foundations
Integral proof foundations
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Change of variables foundations: Jordan covers and two-sided cell distortion
Derive finite-cover boundary estimates and uniform two-sided volumes for small transformed cubes.
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C1 change of variables: a finite-dimensional Jordan/Riemann proof
Prove the nonlinear determinant formula for compact Jordan regions and bounded Riemann-integrable functions, with coverage and boundary hypotheses explicit.
Chapter 16
Gradient descent
Gradient descent
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Gradient Descent: Direction, Step Size and Convergence
Gradient Descent: Direction, Step Size and Convergence
Chapter 17
Complete spectral methods
Complete spectral methods
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Symmetry gives orthogonal eigenvectors
4.3 Symmetry gives orthogonal eigenvectors
Chapter 18
Hypothetical valuation and budgets
Hypothetical valuation and budgets
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Budget-constrained quadratic models
Budget-constrained quadratic models
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Representation-independent valuation of hypothetical outputs
Representation-independent valuation of hypothetical outputs
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Positive-definite quadratic criteria with an equality budget
Positive-definite quadratic criteria with an equality budget