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1. Demonstrate understanding of concepts in linear algebra, relating to vector spaces, linear transformations, orthogonality, eigenvalues and eigenvectors and diagonalisation.
2. Demonstrate understanding of concepts in calculus, relating to differential equations, sequences, series and convergence and multivariable calculus.
3. Employ methods related to these concepts in a variety of applications.
4. Apply logical thinking to problem-solving in context.
5. Demonstrate an understanding of the role of proof in mathematics.
6. Use appropriate technology to aid problem-solving.
7. Demonstrate skills in writing mathematics.

1. Poole, D., Linear Algebra: a Modern Introduction 4th edition (Cengage Learning)
2. Stewart, J., Calculus 8th edition (metric version) (Cengage Learning)

This course uses MyUni extensively and exclusively for providing electronic resources, such as lecture notes, assignment and tutorial questions, and worked solutions. Students should make appropriate use of these resources. MyUni can be accessed here:

This course also makes use of online assessment software for mathematics called Maple TA, which we use to provide students with instantaneous formative feedback. Further details about using Maple TA will be provided on MyUni.

Students are also reminded that they need to check their University email on a daily basis. Sometimes important and time-critical information might be sent by email and students are expected to have read it. Any problems with accessing or managing student email accounts should be directed to Technology Services.

Mathematics IB Summer is taught double-time compared to Semester 1 or 2. The two topics of algebra and calculus detailed below are taught in parallel, with four lectures a week on each. There are two tutorials per week and two assignments per week, one each on algebra and calculus.

Lecture Outline

Algebra

• Revision: Bases, transpose and dimension (2 lectures)
  
  
• Row space, null space, column space, rank theorem (3 lectures)

• Linear Transformations (5 lectures)
  • Definition and basic properties.
  • Kernel and range.
  • Standard matrix.
  • Dimension theorem.
• Orthogonality, Gram-Schmidt (4 lectures)
  • Inner product and orthogonality.
  • Gram-Schmidt process.
  • Orthogonal projection.
  • Ortogonal transformations and matrices.
• Eigenvalues, eigenvectors and diagonalisation (6 lectures)
  
  • Application: Google PageRank.
  • Eigenvalues and eigenvectors.
  • Properties of eigenvalues.
  • Diagonalisation.
  • Symmetric matrices.
  • Orthogonal diagonalisation.
• Singular value decomposition and applications (3 lectures)

Calculus

• Differential Equations (5 lectures)
  • First order separable equations.
  • Phase lines.
  • First order linear equations.
  • Euler and Runge-Kutta methods.
  • Second order constant coefficient homogenous equations.
  • Second order constant coefficient non-homogenous equations.
• Sequences, Series and Convergence (10 lectures)
  
  • Sequences and applications.
  • Series and applications.
  • Power series, Taylor series.
  • Radius of convergence.
• Multivariable Calculus (7 lectures)
  
  • Surfaces in three dimensions.
  • Functions of several variables including polar coordinates.
  • Limits and continuity in two variables.
  • Partial derivatives.
  • Directional derivatives and the gradient.
  • Extrema of functions of two variables.

Tutorial Outline

Tutorial 1: Algebra: Subspaces, linear independence. Row, column and null space. Calculus: First order DEs. Second order DEs.

Tutorial 2: Algebra: Rank theorem. Distance, angle and orthogonality. Orthogonality. Gram-Schmidt process, projections, linear transformations. Calculus: Logistic equation. Limits, Squeeze Theorem.

Tutorial 3: Algebra: Linear transformations: kernel, range, standard matrix. Composition of linear transformations, conic sections. Calculus: Limits, improper integrals. L'Hopital's rule. Discontinuities.

Tutorial 4: Algebra: Eigenvalues, eigenvectors, diagonalisation. Standard form for conics and quadrics. Calculus: Newton's Method. Intermediate Value Theorem. Max-min problems. Mean Value Theorem.

Tutorial 5: Functions of 2 variables, limits and continuity. Limits, continuity, partial derivatives of functions of 2 variables. Mean Value Theorems, graph sketching. Taylor and Maclaurin polynomials.

Tutorial 6: Tangent planes to surfaces. Chain rule. Directional derivatives. Maximum rate of change. Classification of critical points. Series, convergence. Taylor and Maclaurin series.

Note: precise tutorial content may vary due to the vagaries of public holidays.

Assignments 1-6 are made up of a written component, worth a total of 7.5%, and an online (Maple TA) component, worth 7.5%.

Calculus hand-in assignments will be due Mondays at 2:00pm in EM105.
Calculus Maple TA assignments will be due Mondays at 5:00pm.
Algebra hand-in assignments will be due Thursdays at 2:00pm in EM105.
Algebra Maple TA assignments will be due Thursdays at 5:00pm.

All assessment materials will be available via MyUni.

1. All written assignments are to be submitted at the designated time and place with a signed cover sheet attached.
2. Late assignments will not be accepted without a medical certificate.
3. Written assignments will have a one week turn-around time for feedback to students.
4. Online Maple TA assignments provide instantaneous feedback to students.