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This playlist of videos on YouTube has many examples of working with matrix algebra.

This revision seminar from 2014 covered matrices and linear equations. In particular, the first part covered the sizes of matrices, matrix addition, multiplication and powers, and transposes. (It was originally given for students in Mathematics IM, but this content has been moved to Mathematics IA.)

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This revision seminar from Sem 2 2019 had a section on matrix multiplication and inverses. The link goes directly to the right section, which starts at about 1 hour, 18 mins.

This revision seminar from Semester 2 2022 started with discussion of various ways to view matrix multiplication, as well as elementary matrices and how to find matrix inverses.

This short seminar from 2016 covered elementary matrices.

This playlist of YouTube videos hasÌýfour worked examples on row operations for solving systems of linear equations:

This revision seminar from 2014 covers matrix operations and also using matrices to solve linear equations. The links will take you to the linear equations section, which begins at about 58 mins.Ìý

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This revision seminar from Sem 2 2022 had a section in the middle on the number of solutions to systems of linear equations. The links below will take you directly to the relevant section, which starts at 52m38s.

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YouTube Playlist of many worked examples covering linear independence, span, subspace and basis.

The following is a list of revision seminars given to Mathematics IA students over the years on span, linear independence and subspaces. They are listed in roughly the order you should try them.

This seminar from 2014Ìýcovers 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 2021 covers what vectors and sets are, and how to tell if things are subspaces, and what a basis is.

This seminar from Sem 2 2023 had a section (starting at 1h22m50s) on span and subspaces and basis.

This seminar from Sem 2 2021 began with a section on span, basis and linear dependence.

This seminar from Sem 1 2022 ended with a section about what vectors and sets are, and how to tell if things are subspaces (starting at 1h40m40s).

This seminar from Sem 1 2017 beganÌýwith a section on linear dependence.

This seminar from Sem 2 2022 had a section (starting atÌý1h33m20s) Ìýon how to tell when vectors are linearly dependent.

This seminar from Sem 2 2017 had a section on the idea of subspaces and useful ways to think about them. The links go directly to the right section, which begins at about 1 hour 12 mins.

This seminar, also from Sem 2 2017, had a section on basis. It covered what a basis is and how to find one for various types of subspaces. The links go directly to the right section of the seminar, which begins at about 1 hour 16 mins.

This seminar from Sem 2 2018 also had a section on basis with examples of finding a basis. The links go directly to the right section of the seminar, which begins at about 1 hour 48 mins.

This seminar from Sem 1 2018 had a section on subspaces. It includedÌýwhat a subspace is, how to tell whether something is a subspace based on how it is written, and proving that things are subspaces. The links go directly to the right section of the seminar, which begins at about 1 hour 10Ìýmins.

This seminar from Sem 1 2019 had a section covering the connected ideas of linear combinations, span, subspace and basis.ÌýThe links go directly to the right section of the seminar, which begins at about 58 mins.

This seminar from Sem 1 2020 coveredÌýthe definition of subspaces, including the different ways to write them, how to tell if a set is a subspace or not, and proving that a set is a subspace.

This seminar from Sem 2 2020 covered the idea of span, and how it relates to what vectors are and how to work with them.

YouTube playlist of worked examples for Optimisation and Convex Sets.

This revision seminar from Sem 1 2022 started with a discussion of convex sets, including a proof that a set is convex.

This revision seminar from Sem 2 2012 coveredÌýthe idea of linear optimisation problems, a couple of examples, and a discussion of choosing the number of basic solutions.

This revision seminar from Sem 1 2016 covered optimisation using slack variables.

This revision seminar from Sem 1 2023 had a section in the middle on optimisation with slack variables (starting 42m36s).

This revision seminar from Sem 2 2016 covered formulating optimisation problems mathematically. That is, going from the text of the problem to the equations and inequalities that need to be solved.

YouTube playlist of three worked examples on determinants:

This seminar from Sem 1 2023 started with a section all about determinants.

This seminar given in Sem 2 2017 had its firstÌýsection on determinants. It coveredÌýhow to calculate determinants, how they're related to various other matrix calculations, and how row operations affect them. (Note the end of the seminar mentions eigenvalues, which are no longer in the Maths IA curriculum.)

This is a playlist of three videoÌýworked examples on functions, domains and ranges, and inverse functions.

This short revision seminar was given in Sem 2 2014, and covered domains and ranges, as well as finding inverse functions.

The beginning of this revision seminar from Sem 1 2015 covered domains and ranges specifically for inverse functions. (The end of the seminar is about trig substitution.)

This beginning of this revision seminar from Sem 1 2018 was about the Heaviside function and other piecewise functions.

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The aboveÌýrevision seminar from Sem 1 2018 also had a section on limits, including the various limit laws. The links below go straight to the right section, which starts at about 40 mins.

This seminar from Sem 1 2022 had a section on hyperbolic trig functions, and also odd and even functions. The links go straight to the right section, which starts at about 1 hour 8 mins 28 seconds.

This seminar was given in Sem 2 2017 and had a section whichÌýcovered several examples of limits, starting at about 58 mins. (Note that this video was given to Maths 1B students because the limits topic used to be in Maths 1B, so the other topics in this seminar are not in the Maths 1A curriculum. Note it includes L'Hopital's rule and limits at infinity, which are not part of the Maths 1A curriculum now.)

This revision seminar from Sem 2 2016Ìýwas given to Maths 1B, and covers both the Bisection Method (which is in Maths 1A) and Newton's Method (which is notÌýin Maths 1A).Ìý

This is a playlist of three video worked examples on derivatives.

This seminar from Sem 1 2015 discussed derivatives of inverses and derivatives of integrals.

This seminar from Sem 2 2023 had a section (starting at 1h1m15s) discussing inverse trig functions and their derivatives.

This seminar from Sem 2 2019 had a section on implicit differentiation. The links will take you straight to the right section, which starts at about 51 mins.

This seminar was given in Sem 1 2016 to students in another course, but that content is now part of Maths 1A. The seminar covered several examples of calculating rates in these situations.

This seminar given in Sem 1 2019 began with a section about related rates, and discusses strategies to take the information presented and turn it into equations connecting the rates. (It was given to students in Maths 1M, but still applies to Maths 1A.)

This seminar given in Sem 1 2022 had a section about related rates, giving two classic examples of related rates problems. The links will take you straight to the right section, which starts at about 35 mins.

The followingÌýseminars concern summation notation and upper and lower sums.

This seminar given in Sem 2 2019 began with a section on sum notation.Ìý

This seminar in Sem 1 2016 discussed how sum notation works, the rules for how it interacts with other operations, and some of the special manipulations you can do with it.

This seminar in Sem 2 2012 covered calculating integrals using upper and lower sums.

This seminar in Sem 2 2023 started with discussion and an example of calculating an integral via upper and lower sums.

This seminar in Sem 2 2021 had one example of calculating an integral using upper and lower sums (starting atÌý1h27m56s).

This seminar in Sem 1 2017 had one example on upper and lower sums. (Most of the rest of this seminar covers topics that are no longer part of Maths 1A.) The links below go straight to the right section, which starts at about 1 hour 6 mins.

This seminar in Sem 2 2018 had a section on sum notation and using it for upper and lower sums. The links below go straight to the right section, which starts at about 43 mins.

This is a playlist of many video worked examples on integration:

These three PDF handouts are useful for doing different types of integration:

• Table of derivatives (PDF handout)
• Useful trigonometric identities (PDF handout)
• Techniques of integration (PDF handout)

This PDF document contains many written worked examples of various techniques of integration.

• Techniques of integration worked examples (PDF booklet)

This seminar was given in Sem 1 2021 and covered the techniques of integration (substitution, by parts, trig substitution, partial fractions and upper and lower sums).

This seminar was given in Sem 1 2014 and covered the entire of the techniques of integration topic. (Note that the seminar also includes reduction formulas and numerical integration,Ìýwhich are no longer in Maths 1A.)

This second part of this seminar from Sem 1 2017, wasÌýlive worked examples of integration problems, focussing on how the decision was made on which technique to use. Examples included partial fractions and integration by parts (and also reduction formulas, though this isÌýno longer in the Maths 1A curriculum). The links below go straight to the right section, which begins at about 26 mins.

The following seminars each give several examples of specific techniques of integration.

This seminar from Sem 2 2017 begins by discussing integration by parts, and then trig substitution. In particular, how to choose the substitution and the overall process.

This whole revision seminar from Sem 2 2016 is on trig substitution. (Note the last few minutes did not record, but those few minutes concerned concepts that are not officially in the Maths 1A curriculum anyway.)

This seminar from Sem 2 2018 began with aÌýsection on trig substitution, including recognising when to do it and how to choose the right one, and how to perform it using both trig identities and triangles.

This seminar from Sem 2 2021 has an example of doing trig substitution (starting at about 1h8m27s).

This seminar from Sem 1 2019 began with a section on a difficult trig substitution example.

This seminar from Sem 2 2021 covered why partial fractions works the way it does and did a few examples of using partial fractions to integrate rational functions.

This seminar from Sem 2 2016 covered the integration technique of rewriting using partial fractions.

In Semester 2 2016, the MLC ran a revision seminar on improper integrals. David discussed what they are, and did several examples showing how to decide what to do. To watch the video or download the notes, follow the links below:

In Summer Semester 2017, the MLC ran another revision seminar on improper integrals, basically to go through some more difficult examples than in the previous seminar. To watch the video or download the notes, follow the links below:

The revision seminars from each semester cover various different topics and you may find it easier to watch each one as a whole seminar. They are collected here from 2018 onwards (earlier seminars are only available under the topic headings above).

Sem 2 2023

Calculating integrals with upper and lower sums (at the start), then inverse trig functions and their derivatives (starting at 1h1m15s), then span and subspaces and basis (starting at 1h22m50s).

Sem 1 2023

Determinants (at the start), optimisation with slack variables (starting 42m36s), and subspaces (starting at 1h40m40s).

Sem 2 2022

Matrix multiplication, inverses and elementary matrices (at the start), and also row reducing to find solutions to linear equations (starting at 52m38s), as well as how to tell if vectors are linearly independent or dependent (starting at 1h33m20s).

Sem 1 2022

Convex sets (at the start), related rates (34m58s), hyperbolic trig functions (1h8m27s), and even and odd functions (1h24m47m).

Sem 2 2021

Span, basis and linear independence, then trig substitution (starting at 1h8m27s), then calculating integrals with upper and lower sums (starting at 1h27m56s).

Sem 1 2021

Vectors,Ìýsets and subspaces and bases.

Integration, including techniques and upper and lower sums.

Sem 2 2020

Definitions and calculations about span.

Sem 1 2020

Definitions and proofs about subspaces

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Sem 2 2019

Sum notation,Ìýimplicit differentiation,Ìýmatrix multiplication and matrix inverses.

Sem 1 2019

Difficult trig substitutions, and theÌýconnected ideas of linear combinations, span, subspace and basis.

Sem 2 2018

Trigonometric substitution, upper and lower sums, and basis.

Sem 1 2018

The Heaviside function and other piecewise functions,Ìýlimits and subspaces.