Mathematics IA

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

• Video worked examples: matrix algebra (YouTube Playlist)

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.)

• Revision seminar: matrices and linear equations, 2014 (YouTube) 
• Revision seminar notes: matrices and linear equations, 2014 (PDF)

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.

• Revision seminar section: matrix multiplication and inverses, Sem 2 2019 (YouTube)
• Revision seminar notes: matrix multiplication and inverses, Sem 2 2019 (PDF)

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.

• Revision seminar section: matrix multiplication, elementary matrices and inverses, Sem 2 2022 (YouTube)
• Revision seminar notes: matrix multiplication, elementary matrices and inverses, Sem 2 2022 (PDF)

This short seminar from 2016 covered elementary matrices.

• Revision seminar: Elementary matrices, 2016 (YouTube)
• Revision seminar notes: Elementary matrices, 2016 (PDF)

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

• Maths IA Algebra examples: linear systems and Gauss-Jordan elimination (YouTube)

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. 

• Revision seminar section: linear equations, 2014 (YouTube) 
• Revision seminar notes: matrices and linear equations 2014 (PDF)

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.

• Revision seminar section: linear equations, Sem 2 2022 (YouTube) 
• Revision seminar notes: linear equations, Sem 2 2022 (PDF)

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

• Maths IA algebra examples: spanning sets and linearly independent sets (YouTube)

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 2014 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.)

• Revision seminar: independence, subspaces and basis, 2014 (YouTube)
• Revision seminar notes: independence, subspaces and basis 2014 (PDF)

This seminar from Sem 2 2024 covers span, subspace, basis and linear independence (at the start).

• Maths IA revision seminar Sem 2 2024 (YouTube)
• Document camera notes: span, subspace, basis 2024 (PDF)

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

• Revision seminar: vectors, sets and subspaces Sem 1 2021 (YouTube)
• Revision seminar: vectors, sets and subspaces Sem 1 2021 (Echo360)

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

• Revision seminar section: span and basis Sem 2 2023 (YouTube)
• Revision seminar notes: span and basis 2023 (PDF)

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

• Revision seminar section: span, basis and linear dependence Sem 2 2021 (YouTube)

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).

• Maths IA revision seminar Sem 1 2023 (YouTube)
• Document camera notes: subspaces 2023 (PDF)

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

• Revision seminar section: linear dependence, Sem 1 2017 (YouTube)
• Revision seminar notes: linear dependence Sem 1 2027 (PDF)

This seminar from Sem 2 2022 had a section (starting at 1h33m20s)  on how to tell when vectors are linearly dependent.

• Revision seminar section: linear dependence, Sem 2 2022 (YouTube)
• Revision seminar notes: linear dependence Sem 2 2022 (PDF)

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.

• Revision seminar section: the idea of subspaces, Sem 2 2017 (YouTube)
• Revision seminar notes: the idea of subspaces Sem 2 2017 (PDF)

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.

• Revision seminar section: basis, Sem 2 2017 (YouTube)
• Revision seminar notes: basis, Sem 2 2017 (PDF)

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.

• Revision seminar section: basis, Sem 2 2018 (YouTube)
• Revision seminar notes: basis, Sem 2 2018 (PDF)

This seminar from Sem 1 2018 had a section on subspaces. It included what 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 10 mins.

• Revision seminar section: subspaces, Sem 1 2018 (YouTube)
• Revision seminar notes: subspaces, Sem 1 2018 (PDF)

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

• Revision seminar section: linear combinations, span and basis, Sem 1 2019 (YouTube)
• Revision seminar notes: linear combinatons, span and basis, Sem 1 2019 (PDF)

This seminar from Sem 1 2020 covered the 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.

• Revision seminar: subspaces, Sem 1 2020 (YouTube)

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.

• Revision seminar: span, Sem 2 2020 (YouTube)

YouTube playlist of worked examples for Optimisation and Convex Sets.

• Video worked examples: Optimisation and convex sets (YouTube Playlist)

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

• Revision seminar section: convex sets, Sem 1 2022 (YouTube)
• Revision seminar notes: convex sets, Sem 1 2022 (PDF)

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

• Revision seminar: optimisation, Sem 2 2012 (YouTube)
• Revision seminar notes: optimisation, Sem 2 2012 (PDF)

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

• Revision seminar: optimisation using slack variables, Sem 1 2016 (YouTube)
• Revision seminar notes: optimisation using slack variables, Sem 1 2016 (PDF)

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

• Maths IA revision seminar Sem 1 2023 (YouTube)
• Document camera notes: slack variables 2023 (PDF)

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.

• Revision seminar: formulating optimisation problems, Sem 2 2016 (YouTube)
• Revision seminar notes: formulating optimisation problems, Sem 2 2016 (PDF)

YouTube playlist of three worked examples on determinants:

• Video worked examples: determinants (YouTube playlist)

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

• Maths IA revision seminar Sem 1 2023 (YouTube)
• Document camera notes: determinants 2023 (PDF)

This seminar given in Sem 2 2017 had its first section on determinants. It covered how 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.)

• Revision seminar section: determinants, Sem 2 2017 (YouTube)
• Revision seminar notes: determinants, Sem 2 2017 (PDF)

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

• Worked examples: functions (YouTube playlist)

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

• Revision seminar: domains and ranges and inverses, Sem 2 2014 (YouTube)
• Revision seminar notes: domains and ranges and inverses, Sem 2 2014 (PDF)

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.)

• Revision seminar section: domains and ranges for inverse functions, Sem 1 2015 (YouTube)
• Revision seminar notes: domains and ranges for inverse functions, Sem 1 2015 (PDF)

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

• Revision seminar section: heaviside function (and friends), Sem 1 2018 (YouTube) 
• Revision seminar notes: heaviside function (and friends), Sem 1 2018 (PDF)

The above revision 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.

• Revision seminar section: limits, Sem 1 2018 (YouTube)
• iRevision seminar notes: limits,  Sem  1 2018 (PDF)

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.

• Revision seminar section: hyperbolic trig, odd and even, Sem 1 2022 (YouTube)
• Revision seminar notes: hyperbolic trig, Sem 1 2022 (PDF)
• Revision seminar notes: odd and even functions, Sem 1 2022 (PDF)

This seminar was given in Sem 2 2017 and had a section which covered 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.)

• Revision seminar: limits examples, Sem 2 2017 (YouTube)
• Revision seminar notes: limits examples, Sem 2  2017 (PDF)

This revision seminar from Sem 2 2016 was given to Maths 1B, and covers both the Bisection Method (which is in Maths 1A) and Newton's Method (which is not in Maths 1A). 

• Revision seminar: bisection method, Sem 2 2016 (YouTube)
• Revision seminar notes: bisection method, Sem 2 2016 (PDF)

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

• Video worked examples: differentiation (YouTube playlist)

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

• Revision seminar: differentiation of inverses and integrals, Sem 1 2015 (YouTube)
• Revision seminar notes: differentiation of inverses and integrals, Sem 1 2015 (PDF)

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

• Revision seminar section: inverse trig derivatives Sem 2 2023 (YouTube)
• Revision seminar notes: inverse trig derivatives 2023 (PDF)

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.

• Revision seminar section: implicit differentiation, Sem 2 2019 (YouTube)
• Revision seminar notes: implicit differentiation, Sem 2 2019 (PDF)

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.

• Revision seminar: related rates, Sem 1 2016 (YouTube)
• Revision seminar: related rates, Sem 1 2016 (PDF)

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.)

• Revision seminar section: related rates, Sem 1 2019 (YouTube)
• Revision seminar notes: related rates, Sem 1 2019 (PDF)

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.

• Revision seminar section: related rates, Sem 1 2022 (YouTube)
• Revision seminar notes: related rates, Sem 1 2022 (PDF)

The following seminars concern summation notation and upper and lower sums.

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

• Revision seminar section: sum notation, Sem 1 2019 (YouTube)
• Revision seminar notes: sum notation, Sem 1 2019 (PDF)

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.

• Revision seminar: sum notation, Sem 1 2016 (YouTube)
• Revision seminar notes: sum notation, Sem 1 2016 (PDF)

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

• Revision seminar: upper and lower sums, Sem 2 2012 (YouTube)
• Revision seminar notes: upper and lower sums, Sem 2 2012 (PDF)

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

• Revision seminar section: upper and lower sums Sem 2 2023 (YouTube)
• Revision seminar notes: upper and lower sums 2023 (PDF)

This seminar in Sem 2 2021 had one example of calculating an integral using upper and lower sums (starting at 1h27m56s).

• Revision seminar section: upper and lower sums Sem 2 2021 (YouTube)

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.

• Revision seminar section: upper and lower sums example, Sem 1 2017 (YouTube)
• Revision seminar notes: upper and lower sums example, Sem 1 2017 (PDF)

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.

• Revision seminar section: sums and upper and lower sums, Sem 2 2018 (YouTube)
• Revision seminar notes: sums and upper and lower sums, Sem 2 2018 (PDF)

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

• Video worked examples: integration (YouTube playlist)

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).

• Revision seminar: Integration Sem 1 2021 (YouTube)

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, which are no longer in Maths 1A.)

• Revision seminar: techniques of integration, Sem 1 2014 (YouTube)
• Revision seminar notes: techniques of integration, Sem 1 2014 (PDF)

This second part of this seminar from Sem 1 2017, was live 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 is no longer in the Maths 1A curriculum). The links below go straight to the right section, which begins at about 26 mins.

• Revision seminar section: integration examples, Sem 1 2017 (YouTube)
• Revision seminar notes: integration examples, Sem 1 2017 (PDF)

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.

• Revision seminar section: integration by parts and trig substiution, Sem 2 2017 (YouTube)
• Revision seminar notes: integration by parts, Sem 2 2017 (PDF)
• Revision seminar notes: trig substitution, Sem 2 2017 (PDF)

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.)

• Revision seminar: trig substitution, Sem 2 2016 (YouTube)
• Revision seminar notes: trig substitution, Sem 2 2016 (PDF)

This seminar from Sem 2 2018 began with a section 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.

• Revision seminar section: trig substitution, Sem 2 2018 (YouTube)

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

• Revision seminar section: trig substitution Sem 2 2021 (YouTube)

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

• Revision seminar section: trig substitution, Sem 1 2019 (YouTube)
• Revision seminar notes: trig substitution, Sem 1 2019 (PDF)

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.

• Revision seminar: partial fractions, Sem 2 2021 (YouTube)
• Revision seminar: partial fractions, Sem 2 2021 (Echo360)

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

• Revision seminar: partial fractions, Sem 2 2016 (YouTube)
• Revision seminar notes: partial fractions, Sem 2 2016 (PDF)

In Semester 2 2024, David ran a revision seminar where the second part discussed span the fancy integration techniques of trig substitution and rewriting with partial fractions (starting at 1h27m35s).

• Maths IA revision seminar section Sem 2 2024 (YouTube)
• Document camera notes: fancy integration 2024 (PDF)

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:

• Revision seminar: Improper integrals 2016 (YouTube)
• Revision seminar notes: Improper integrals 2016 (PDF)

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:

• Revision seminar: Improper integrals 2017 (YouTube)
• Revision seminar notes: Improper integrals 2017 (PDF)

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 2024

David discussed span, subspace, basis and linear independence (at the start), and then discussed the fancy integration techniques of trig substitution and rewriting with partial fractions (starting at 1h27m35s).

• Maths IA revision seminar Sem 2 2024 (YouTube)
• Document camera notes: span, subspace, basis 2024 (PDF)
• Document camera notes: fancy integration 2024 (PDF)

Sem 1 2024

Improper integrals, then a very short discussion of strategies for dealing with application problems, and some comments on the fundamental theorem of calculus.

• Maths IA revision seminar Sem 1 2024 (YouTube)
• Document camera notes: improper integrals and other thoughts 2024 (PDF)

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).

• Maths IA revision seminar Sem 2 2023 (YouTube)
• Document camera notes: upper and lower sums 2023 (PDF)
• Document camera notes: inverse trig functions 2023 (PDF)
• Document camera notes: span and basis 2023 (PDF)

Sem 1 2023

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

• Maths IA revision seminar Sem 1 2023 (YouTube)
• Document camera notes: determinants 2023 (PDF)
• Document camera notes: slack variables 2023 (PDF)
• Document camera notes: subspaces 2023 (PDF)

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).

• Maths 1A revision seminar Sem 2 2022 (YouTube)
• Document camera notes: matrices (PDF)
• Document camera notes: solutions to linear equations (PDF)
• Document camera notes: linear independence (PDF)

Sem 1 2022

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

• Maths 1A revision seminar Sem 1 2022 (YouTube)
• Document camera notes: convex sets (PDF)
• Document camera notes: related rates (PDF)
• Document camera notes: hyperbolic trig (PDF)
• Document camera notes: even and odd functions (PDF)

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).

• Maths 1A revision seminar Sem 2 2021 (YouTube)

Sem 1 2021

Vectors, sets and subspaces and bases.

• Maths 1A first revision seminar Sem 1 2021 (YouTube)

Integration, including techniques and upper and lower sums.

• Maths 1A second revision seminar Sem 1 2021 (YouTube)

Sem 2 2020

Definitions and calculations about span.

• Maths 1A revision seminar Sem 2 2020 (YouTube)

Sem 1 2020

Definitions and proofs about subspaces

• Maths 1A revision seminar Sem 1 2020 (YouTube)

Sem 2 2019

Sum notation, implicit differentiation, matrix multiplication and matrix inverses.

• Maths IA revision seminar Sem 2 2019 (YouTube)
• Maths IA revision seminar notes: sum notation Sem 2 2019 (PDF)
• Maths 1A revision seminar notes: implicit differentiation Sem 2 2019 (PDF)
• Maths 1A revision seminar notes: matrix multiplication and inverses Sem 2 2019 (PDF)

Sem 1 2019

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

• Maths IA revision seminar Sem 1 2019 (YouTube)
• Maths IA revision seminar notes: trig substitutions Sem 1 2019 (PDF)
• Maths IA revision seminar notes: span and basis Sem 1 2019 (PDF)

Sem 2 2018

Trigonometric substitution, upper and lower sums, and basis.

• Maths IA revision seminar Sem 2 2018 (YouTube)
• Maths IA revision seminar notes: trig substitution Sem 2 2018 (PDF)
• Maths IA revision seminar notes: upper and lower sums Sem 2 2018 (PDF)
• Maths IA revision seminar notes: basis Sem 2 2018 (PDF)

Sem 1 2018

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

• Maths IA revision seminar Sem 1 2018 (YouTube)
• Maths IA revision seminar notes: heaviside function and friends Sem 1 2018 (PDF)
• Maths IA revision seminar notes: limits Sem 1 2018 (PDF)
• Maths IA revision seminar notes: subspaces Sem 1 2018 (PDF)