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This seminar was given to Maths IA students in 2014 and covers all of the concepts to do with independence, span, subspaces and basis, beginning with what vectors and sets are. (David considers this one of his best revision seminars ever.)

This seminar from 2012 covers linear transformations, focusing especially on kernel and range of linear transformations (unfortunately the last several minutes were cut off from this video).

This seminar in Semester 1 2018 started with a section on kernels and ranges for linear transformations.

This seminar from Semester 2 2017 started with a section covering various thoughts about the standard matrices of linear transformations.

This seminar from Semester 2 2020 had a section on the subspaces associated with matrices and linear transformations: rowspace, columnspace, nullspace, kernel and range (starting at 21m40s).

This seminar from Summer Semester 2019 had a section on orthogonal projection (starting at 57m).

This revision seminar from Summer Semester 2019 had a section on orthogonal projection (starting at 57m).

This revision seminar from 2016 coveredÌýorthonormal basis and the Gram-Schmidt process in linear algebra.

This revision seminar from Semester 2 2021 had a section on orthogonal projection and the Gram-Schmidt process (starting atÌý1h59m).

This revision seminar from Semester 2 2022 started with a section on orthonormal bases. (Apologies for the commentary running behind the audio from a couple of students. I couldn't remove it.)

This PDF handout list various facts about eigenvalues and some examples of classic problems using them.

• Facts about eigenvalues handout

This seminar from 2012 covers eigenvalues and eigenvectors for matrices (it was given for students in Maths 1A but this topic has now been moved to Maths 1B).

This seminar from Summer Semester 2020 started with a section on eigenvectors.

This seminar from Semester 1 2018 had a section on guessing eigenvectors from drawings of transformations (at 1h19m24s).

This seminar in Sem 1 2017 had a section on Markov chains starting at 23m30s.Ìý(This was given for Maths 1A students but this topic is now in Maths 1B.)

This seminar from Summer Semester 2023 had a section on diagonalisation starting at 1h40s.

This seminar from Semester 2 2017 began with a section onÌýorthogonal diagonalisation.

This seminar from Semester 2 2019 had a section onÌýsingular value decomposition (starting at 1h12m50s).

This seminar from Semester 2 2020 started with a section on orthogonal matrices, eigenvalues and the trace.

This handout lists advice you can give yourself when you are solving problems, especially proofs. (This is the latest version -- the old version is shown in the 2014 seminar below.)

• Solving problems handout

This seminar from Summer 2014 givesÌýadvice for creating proofs in linear algebra involving subspaces, linear transformations and matrices.

This seminar from Summer Semester 2021 has a section on how to come up with proofs in Algebra (starting at 28m30s).

This seminar from 2014 covered various topics on differential equations, including first order separable equations, integrating factors, second order linear equations and the logistic equation. In this seminar, there was a Prezi presentation showing the process of solving linear second order differential equations. There is a link below to a PDF handout version.

• Second-order linear differential equations with constant coefficients handout (PDF)Ìý

In this seminar from Semester 2 2018, the first part covered phase diagrams for differential equations. There was a Desmos graph that David showed in this seminar, which there is a link to below.

This seminar from Semester 1 2023Ìýdiscussed second order linear (constant coefficient) differential equations, including finding the general solution to homogeneous equations via the characteristic equation, and finding a particular solution to a non-homogeneous equation.

This seminar from Summer Semester 2023 had a section on particular solutions to second order DEs starting at 19m34s.

This seminar from Semester 1 2019 started with a section on numerical solutions for differential equations using Euler's method.

In almost every semester when the MLC runs revision seminars, students request to discuss Taylor series or other series. So there are very manyÌýseminars with sections on series. I have tried to list them in the order that I think is most useful.Ìý

This handout lists the various tests for convergence, as well as showing the process of finding an interval of convergence.

• Tests for convergence handout

This seminar in Summer Semester 2019 gave an intro into what Taylor series and Taylor polynomials are, then gave several examples of finding them and working with the error formula.

This seminar in Semester 2 2023 discussed various aspects of Taylor series and Taylor polynomials.

This seminar in Semester 2 2019 started with a section onÌýTaylor series.

This seminar in Semester 2Ìý2017 ended with a section which showed an overview of infinite series and Taylor series (starting at 1h21m).

This seminar from Semester 2 2021 began with a big section on sequences and series and Taylor series.

Also in Semester 2 2021, David recorded some worked examples on problems involving infinite series, power series and Taylor series.

This seminar from Semester 1 2022 ended with a section on series and Taylor series. (starting at 1h13m44s).

This seminar from Semester 2 2022 had a section on sequences (starting at 1h26m20s).Ìý

This seminar from 2015 coveredÌýTaylor series and how to find Taylor series for some functions by making them out of other functions.

This seminar from Semester 1 2019 ended with a section withÌývarious aspects of power series (starts at 1h29m51s).

This revision seminar from 2016 coveredÌýworking with the error formula for Taylor polynomials.

This revision seminar from Semester 1 2020 showedÌývarious questions about Taylor series errors/remainders.

This revision seminar in Summer Semester 2018 covered Taylor/Maclaurin polynomials, using Taylor's error formula and a bit of infinite series.

This seminar in Semester 2 2018 was recorded in two parts. At the end of the first part (at 32m38s), David started talking about series, and then continued in the second part.

This seminar from 2015 coveredÌýmost of the topic of infinite series, including tests for convergence and intervals of convergence.

This seminar from 2015 coveredÌýintervals of convergence and the common Maclaurin series, especially the binomial series.

This seminar in Summer Semester 2022 started with a section on intervals of convergence.

This seminar in Summer Semester 2019 ended with a section on infinite series examples (starting at 1h30m).

This seminar from Summer Semester 2020 ended with a section onÌýthe binomial series (starting at 1h59m39s).

This seminar from Summer Semester 2021 ended with a section on sequences and series (starting at 1h13m22s).

This seminar from Semester 1 2019 has a section on numerical solutions for equations using the Newton-Raphson method (startingÌýat 25m53s).

This revision seminar from 2016 coveredÌýthe use of Newton's method (and the bisection method) to find solutions of equations.

This same seminar from Semester 1Ìý2019 also includes a section on L'Hopital's rule (starting at 1h30s).

This seminar from Semester 2 2022 has a section on L'Hopital's Rule (starting at 54m38s). (Apologies for the commentary running behind the audio from a couple of students. I couldn't remove it.)

This seminar from Summer Semester 2023 started withÌýan example of calculating limits at infinity.

In this seminar from Summer Semester 2019,ÌýDavid spent a lot of time talking about different ways to imagine multivariable functions and their derivatives.

This seminar from Summer Semester 2019 began with a section onÌýcritical points of multivariable functions.

This seminar from 2012 covers domains, ranges, partial derivatives, maxima and minima in multivariable calculus..

This seminar from 2013 covered directional derivatives and tangent planes in multivariable (3D) calculus.

This seminar from Semester 1 2021 covered multivariable derivatives, including directional derivatives, partial derivatives and critical points.

This seminar from Semester 1 2022 began with a section on two-variable functions including ways to define a surface, finding and classifying critical points, and tangent planes.

This revision seminar from 2015 coveredÌýthe chain rule in multivariable calculus.

This seminar from Summer Semester 2021 began with a section on the multivariable chain rule.

This seminar from Summer Semester 2022 ended with a section on directional derivatives (starting at 1h35m).

Ìý

These are the MLC revision seminars for Maths 1B that have been done since 2018, with the newest seminars at the top.

2024

Semester 2 2024: Nicholas did various problems on the following topics: Eigenvalues (at the start), Gramm-Schmidt Process (at 26m58s), Kernel and range of linear transformations (at 35m58s), Multivariable chain rule (at 59m55s), Tangent planes (at 1h10m53s), Taylor series (at 1h17m7s).

Semester 1 2024:ÌýNicholas discussed power series and Taylor series (at the start), subspaces (at 52m7s), orthogonal projection (at 1h24m10s), and column space, row space and null space (at 1h38m36s).

Summer Semester 2024: Nicholas discussedÌýmiscellaneous topics, including limits, tangent planes of surfaces in 3D, critical points of two-variable functions, orthogonal diagonalisation, and the Gram-Schmidt process.

2023

Semester 2 2023:ÌýDavid discussed various aspects of Taylor series and Taylor polynomials.

Semester 1 2023: David discussed second order linear (constant coefficient) differential equations, including finding the general solution to homogeneous equations via the characteristic equation, and finding a particular solution to a non-homogeneous equation.

Summer Semester 2023: David discussed limits at infinity (at the start), particular solutions to second order DEs (starting at 19m34s), Big O notation (starting at 50m10s), and diagonalisation (starting at 1h40s).

2022

Semester 2 2022:ÌýDavid discussed orthonormal bases (at the start), L'Hopital's Rule (starting at 54m38s), and Sequences (starting at 1h26m20s). (Apologies for the commentary running behind the audio from a couple of students. I couldn't remove it.)

Semester 1 2022: David discussedÌýtwo-variable functions including ways to define a surface, finding and classifying critical points, and tangent planes, and then series convergence (at 1h13m44s) and a little on Taylor series (at 1h50m50s).

Summer Semester 2022: David discussed intervals of convergence, and thenÌý(starting at 1h35m) he talked about directional derivatives.

2021

Semester 2 20201: David discussed sequences and series, and briefly talked about big O notation (at 1h46m) and orthogonal projection and the Gram-Schmidt process (at 1h59m).

Also in Semester 2 2021, David recorded several worked examples of problems involving series and Taylor series:

Semester 1 2021:ÌýDavid discussed multivariable derivatives, including directional derivatives, partial derivatives and critical points.

Summer Semester 2021:ÌýDavid discussed the multivariable chain rule, how to come up with proofs in Algebra (starting at 28m30s), and sequences and series (starting at 1 hour 13 mins 22sec).

2020

Semester 2 2020:ÌýDavid discussed orthogonal matrices, eigenvalues and the trace, and then a large section (starting at 21m40s) on the subspaces associated with matrices and with linear transformations, including row space, column space, null space, range and kernel.

Semester 1 2020:ÌýDavid did various questions about Taylor series errors/remainders.

Summer Semester 2020:ÌýDavid discussedÌýeigenvectors, markov chains (starting at 1h14m30s ), and the binomial series (starting at 1h59m39s).

2019

Semester 2 2019: David discussed Taylor series (54s) and singular value decomposition (1h12m50s).

Semester 1 2019:ÌýDavid discussed numerical solutions for differential equations using Euler's method; numerical solutions for equations using the Newton-Raphson method (starts at 25m53s); L'Hopital's Rule (starts at 1h30s); various aspects of power series (starts at 1h29m51s).

Summer Semester 2019 (3rd seminar):ÌýDavid covered critical points of multivariable functions,Ìýorthogonal projection (starting at 57m), and some infinite series examples (starting at 1h30m).

Summer Semester 2019 (2nd seminar):ÌýDavid gave an intro into what Taylor series and Taylor polynomials are, then gave several examples of finding them and working with the error formula.

Summer Semester 2019 (1st seminar):ÌýDavid spent a lot of time talking about different ways to imagine multivariable functions and their derivatives.

2018

Semester 2 2018: This seminar was recorded in two different parts because we were forced to move rooms. The first partÌýcovered phase diagrams for differential equations, and started talking about series (at 32m38s).ÌýIn the second part,ÌýDavid continued discussingÌýinfinite series and power series.

Ìý

These are the Maths IB revision seminars from 2012 to 2018, with the newest seminars at the top.

NOTE: The curriculum for Maths IB changed in Semester 2 2018, so these seminars sometimes contain content that is no longer in the Maths IB curriculum. If in doubt, consult your course notes.

2018

Semester 1 2018: This seminarÌýcovered kernels and ranges for linear transformations, doing an integral using a power series (1h4m3s), guessing eigenvectors from drawings of transformations (1h19m24s), multivariable limits (1h40m57s) and the intermediate value theorem (2h8m30s) [the last part of the video was cut off before any examples of the IVT were done].

Summer Semester 2018: This seminar covered Taylor/Maclaurin polynomials, using Taylor's error formula and a bit of infinite series.

2017

Semester 2 2017: This seminar covered orthogonal diagonalisation, and also several examples of limits (starting at 58m25s),Ìýincluding limits at infinity and limits concerned with continuity.

Semester 2Ìý2017: This seminar covered various thoughts about the standard matrices of linear transformations, and an overview of infinite series and Taylor series (starting at 1h21m).

2016

This revision seminar from Semester 2 2016 covered improper integrals.

This revision seminar from Semester 2 2016 coveredÌýworking with the error formula for Taylor polynomials.

This revision seminar from Semester 2 2016 coveredÌýthe use of Newton's method and the bisection method to find solutions of equations.

This revision seminar from Summer Semester 2016 coveredÌýorthonormal basis and the Gram-Schmidt process in linear algebra.

2015

This seminar from Semester 2 2015 was recorded in two separate videos. The first video covers Taylor series and how to find Taylor series for some functions by making them out of other functions. The second video covers finding Taylor series and Taylor series errors.

This revision seminar from Summer 2015 coveredÌýthe chain rule in multivariable calculus.

This seminar from Summer 2015 coveredÌýintervals of convergence and the common Maclaurin series, especially the binomial series.

This seminar from Semester 1 2015 coveredÌýmost of the topic of infinite series, including tests for convergence and intervals of convergence.

2014

This seminar from Semester 2 2014 covered various topics on differential equations, including first order separable equations, integrating factors, second order linear equations and the logistic equation.

This seminar from Summer Semester 2014 givesÌýadvice for creating proofs in linear algebra involving subspaces, linear transformations and matrices.

2013

This seminar from Summer Semester 2013 covered directional derivatives and tangent planes in multivariable (3D) calculus.

2012

This seminar fromÌý Semester 2 2012 covers domains, ranges, partial derivatives, maxima and minima in multivariable calculus..

This seminar from Semester 2 2012 covers linear transformations, focusing especially on kernel and range of linear transformations (unfortunately the last several minutes were cut off from this video).

This seminar from Semester 1 2012 covers eigenvalues and eigenvectors for matrices (it was given for students in Maths 1A but this topic has now been moved to Maths 1B).