A-level Further Mathematics for Year 12 - Course 1: Complex Numbers, Matrices, Roots of Polynomial Equations and Vectors from edX

This course by Imperial College London is designed to help you develop the skills you need to succeed in your A-level further maths exams.

You will investigate key topic areas to gain a deeper understanding of the skills and techniques that you can apply throughout your A-level study. These skills include:

Fluency – selecting and applying correct methods to answer with speed and efficiency
Confidence – critically assessing mathematical methods and investigating ways to apply them
Problem-solving – analysing the ‘unfamiliar’ and identifying which skills and techniques you require to answer questions
Constructing mathematical argument – using mathematical tools such as diagrams, graphs, logical deduction, mathematical symbols, mathematical language, construct mathematical argument and present precisely to others
Deep reasoning – analysing and critiquing mathematical techniques, arguments, formulae and proofs to comprehend how they can be applied

Over eight modules, you will be introduced to

complex numbers, their modulus and argument and how they can be represented diagrammatically
matrices, their order, determinant and inverse and their application to linear transformation
roots of polynomial equations and their relationship to coefficients
series, partial fractions and the method of differences
vectors, their scalar produce and how they can be used to define straight lines and planes in 2 and 3 dimensions.

Your initial skillset will be extended to give a clear understanding of how background knowledge underpins the A-level further mathematics course. You’ll also be encouraged to consider how what you know fits into the wider mathematical world.

What you'll learn

How to extend the number system to include and the definition of a complex number.

How to add, subtract, multiply and divide complex numbers.

How to represent complex numbers on an Argand diagram and the modulus and argument of a complex number.

How to write complex numbers in modulus-argument form.

How to define loci in the complex plane.

How to define a matrix by its order.

How to add and subtract conformable matrices.

How to multiply two conformable matrices.

How to use matrices to define linear transformations.

How to find invariant lines and lines of invariant points.

How to find the determinant and inverse of a 2 x 2 and 3 x 3 matrix.

How to use matrices to solve systems of linear equations.

How to use standard series formulae to find the sums of other series.

How to separate algebraic fractions into partial fractions.

How to use the method of differences to find the sum of a series.

How to find the scalar (dot) product of two vectors.

How to define the equation of a line using vectors.

How to define a plane using vectors.

How to use vectors to solve problems involving lines and planes.

What's inside

Syllabus

Module 1: Complex Numbers 1: An Introduction to Complex Numbers

The definition of an imaginary number

The definition of a complex number

Solving simple quadratic equations

Addition, subtraction and multiplication of complex numbers

Complex conjugates and division of complex numbers

Radian measure

Representing complex numbers on the Argand diagram

Module 2: Matrices 1: An Introduction to Matrices

The order of a matrix

Addition and subtraction of conformable matrices

Matrix multiplication

The identity matrix

Matrix transformations in 2 and 3 dimensions

Invariant lines and lines of invariant points

Module 3: Further Algebra and Functions 1: Roots of Polynomial Equations

Solving polynomial equations with real coefficients

The relationship between roots and coefficients in a polynomial equation

Forming a polynomial equation whose roots are a linear transformation of the roots of another polynomial equation

Module 4: Complex Numbers 2: Modulus-Argument form and Loci

The modulus and argument of a complex number

Writing complex numbers in modulus argument form

The geometrical effect of multiplying by a complex number.

Loci on the Argand diagram

Module 5: Matrices 2: Determinants and Inverse Matrices

The determinant of a

square matrix.

The inverse of a

square matrix

Using matrices to solve simultaneous equations (5)

The geometrical interpretation of the solution of a system of equations

Module 6: Further Algebra and Functions 2: Series, Partial Fractions and the Method of Differences

Deriving formulae for series using standard formulae

Separating algebraic fractions into partial fractions

The method of differences

Partial fractions and method of differences

Module 7: Vectors 1: The Scalar (dot) Product and Vector Equations of Lines

The scalar product of two vectors

The vector and Cartesian forms of an equation of a straight line in 2 and 3 dimensions

Solving geometrical problems using vector equations of lines

The dot product and the angle between two lines

Module 8: Vectors 2: The Vector Equations of a Plane and Geometrical Problems with Lines and Planes

The vector and Cartesian forms of the equation of a plane

The vector equation of a plane

Solving geometrical problems with lines and planes using vectors

The intersection of a line and a plane

Perpendicular distance from a point to a plane

Good to know

Know what's good

, what to watch for

, and possible dealbreakers

Develops fluency, confidence, problem-solving, constructing mathematical argument, and deep reasoning in mathematics, which are core skills for A-level studies

Taught by Philip Ramsden, an Instructor at the Department of Mathematics at Imperial College London

Taught by Phil Chaffe, the Senior Lecturer at the Department of Mathematics at Imperial College London

Examines complex numbers, their modulus and argument, and how they can be represented diagrammatically, topics highly relevant to A-level mathematics

Develops matrices, their order, determinant, inverse, and their application to linear transformation, core aspects of A-level mathematics

Strengthens an existing foundation for A-level mathematics by introducing roots of polynomial equations and their relationship to coefficients

Provides a comprehensive study of series, partial fractions, and the method of differences, concepts foundational in A-level mathematics

Builds a strong foundation for A-level mathematics by introducing vectors, their scalar product, and how they can be used to define straight lines and planes

Explicitly requires learners to come in with background knowledge, which may be a barrier for some learners

Activities

Be better prepared before your course. Deepen your understanding during and after it. Supplement your coursework and achieve mastery of the topics covered in A-level Further Mathematics for Year 12 - Course 1: Complex Numbers, Matrices, Roots of Polynomial Equations and Vectors with these activities:

Review matrices

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Helps review matrix operations and concepts for better understanding.

Browse courses on Matrices

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Revisit notes on matrix addition and subtraction
Practice multiplying matrices
Review matrix determinants and their properties

Solve practice problems on complex numbers

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Enhances fluency in manipulating and solving problems involving complex numbers.

Browse courses on Complex Numbers

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Find practice problems on complex number operations
Attempt to solve the problems independently
Check solutions and identify areas for improvement

Read 'Linear Algebra and its Applications' by Gilbert Strang

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Provides a comprehensive overview of linear algebra concepts, including matrices.

View Linear Algebra for Everyone on Amazon

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Read Chapter 1: Introduction to Linear Algebra
Focus on sections covering matrices and their operations
Complete practice problems at the end of each section

Six other activities

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Show all nine activities

Collect resources on complex numbers and matrices

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Provides easy access to a variety of materials for further exploration and reinforcement.

Browse courses on Complex Numbers

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Search for online resources, articles, and videos on complex numbers and matrices
Organize and categorize the resources based on topic and difficulty level
Share the compilation with classmates for collaborative learning

Join a study group for problem-solving

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Facilitates collaboration, peer learning, and diverse perspectives on problem-solving.

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Identify classmates interested in forming a study group
Establish regular meeting times and a shared study space
Work together on practice problems, discuss concepts, and support each other

Follow tutorials on matrix transformations

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Reinforces understanding of how matrices are used to transform vectors and points.

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Search for tutorials on matrix transformations
Follow along with the steps and examples provided
Attempt practice exercises to solidify concepts

Create a cheat sheet on vector operations

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Summarizes key vector operations and their properties for easy reference.

Browse courses on Vector Operations

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Gather information on vector addition, subtraction, and dot product
Organize and present the information in a concise and visually appealing manner
Use the cheat sheet as a quick reference during coursework

Tutor students in mathematics

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Enhances understanding by explaining concepts to others and solidifies knowledge through repetition.

Browse courses on Mathematics

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Contact local schools or tutoring centers to offer your services
Identify students who need assistance in mathematics
Provide personalized tutoring sessions, focusing on areas where students struggle

Contribute to open-source projects related to linear algebra

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Provides practical experience in applying linear algebra concepts and contributing to a community of developers.

Browse courses on Linear Algebra

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Identify open-source projects that utilize linear algebra
Join the project community and explore their codebase
Identify areas where you can contribute your knowledge and skills

Career center

Learners who complete A-level Further Mathematics for Year 12 - Course 1: Complex Numbers, Matrices, Roots of Polynomial Equations and Vectors will develop knowledge and skills that may be useful to these careers:

Mathematician

Mathematicians study the properties and relationships of numbers, quantities, and structures. These professionals use their knowledge to solve problems, develop new theories, and advance our understanding of the world around us. This course can help build a strong foundation in the fundamental concepts of mathematics, such as algebra, calculus, and geometry, which are essential for success as a Mathematician.

See salaries and explore the career path for Mathematician

Data Scientist

Data Scientists use their skills in mathematics, statistics, and computer programming to extract insights from data. These professionals work in a variety of industries, helping organizations make informed decisions based on data-driven analysis. This course can help build a strong foundation in the mathematical and statistical concepts that are essential for success as a Data Scientist, including probability, statistics, and linear algebra.

See salaries and explore the career path for Data Scientist

Quantitative Analyst

Quantitative Analysts use mathematical and statistical models to analyze financial data and make investment decisions. These professionals play a critical role in the financial industry, helping investment firms and hedge funds make informed decisions about where to allocate their capital. To succeed in this role, professionals should have a strong understanding of mathematics, statistics, and computer programming. This course can help build a solid foundation in these key areas, providing a valuable advantage for those aspiring to become Quantitative Analysts.

See salaries and explore the career path for Quantitative Analyst

Actuary

Actuaries analyze financial data to determine the likelihood of future events, such as death, disability, or property damage. These professionals use their mathematical skills to develop and price insurance policies, as well as to manage risk for companies and individuals. This course can help build a foundation in essential mathematical skills, such as probability, statistics, and calculus, which are crucial for success as an Actuary.

See salaries and explore the career path for Actuary

Statistician

Statisticians collect, analyze, and interpret data to provide insights and make predictions. This information is used by a variety of organizations to make informed decisions. To succeed as a Statistician, professionals should have a strong foundation in mathematics, statistics, and probability. This course can help build a solid foundation in these key areas, providing a valuable advantage for those aspiring to become Statisticians.

See salaries and explore the career path for Statistician

Operations Research Analyst

Operations Research Analysts use mathematical and analytical techniques to solve complex problems in business and industry. These professionals help organizations optimize their operations, reduce costs, and improve efficiency. To succeed in this role, professionals should have a strong understanding of mathematics, statistics, and optimization techniques. This course can help build a solid foundation in these key areas, preparing individuals for success as Operations Research Analysts.

See salaries and explore the career path for Operations Research Analyst

Machine Learning Engineer

Machine Learning Engineers design and implement machine learning algorithms to solve complex problems in a variety of industries. These professionals use their mathematical and programming skills to develop AI-powered solutions that can automate tasks, improve decision-making, and provide insights from data. This course can help build a solid foundation in the mathematical concepts underlying machine learning, such as linear algebra, calculus, and probability, which are essential for success as a Machine Learning Engineer.

See salaries and explore the career path for Machine Learning Engineer

Data Analyst

Data Analysts collect, clean, and analyze data to identify trends and patterns. This information is then used by businesses to make informed decisions. To succeed in this role, professionals should have a strong understanding of mathematics, statistics, and computer programming. The mathematical concepts covered in this course, particularly those related to data analysis and probability, can provide a strong foundation for a career as a Data Analyst.

See salaries and explore the career path for Data Analyst

Economist

Economists study how societies produce, distribute, and consume goods and services. They use mathematical models and statistical analysis to understand economic trends and make predictions. To succeed in this role, professionals should have a strong understanding of mathematics, statistics, and economics. This course can equip individuals with a solid foundation in mathematical concepts relevant to economic modeling, such as calculus, linear algebra, and differential equations, which are essential for success as an Economist.

See salaries and explore the career path for Economist

Physics

Physicists study the fundamental laws of nature and the behavior of matter and energy. These professionals use their knowledge to develop new technologies and advance our understanding of the universe. While this course does not directly teach physics, the mathematical concepts covered, such as calculus, linear algebra, and differential equations, are essential for success as a Physicist. A strong foundation in these mathematical concepts can help Physics students excel in their studies and prepare for a successful career in the field.

See salaries and explore the career path for Physics

Financial Analyst

Financial Analysts help companies, organizations, and individuals evaluate and manage their financial resources and investments. To succeed as a Financial Analyst, professionals should have a strong understanding of financial markets, economics, and mathematics. This course can help build a foundation in mathematics relevant to financial modeling, statistics, and probability, all of which are important for success in this role.

See salaries and explore the career path for Financial Analyst

Engineer

Engineers design, develop, and maintain machines, structures, and systems. These professionals use their knowledge of mathematics, science, and engineering principles to solve problems and create solutions that meet the needs of society. While this course does not directly teach engineering, the mathematical concepts covered, such as calculus, linear algebra, and differential equations, are essential for success as an Engineer. A strong foundation in these mathematical concepts can help Engineering students excel in their studies and prepare for a successful career in the field.

See salaries and explore the career path for Engineer

Teacher

Teachers educate and inspire students in a variety of subjects, including mathematics. To succeed in this role, professionals should have a strong understanding of the subject matter they teach, as well as effective teaching methods. This course can help build a strong foundation in the mathematical concepts covered in A-level Further Mathematics, which can be beneficial for those aspiring to become Mathematics teachers. It can also provide insights into effective teaching strategies and lesson planning.

See salaries and explore the career path for Teacher

Software Engineer

Software Engineers design, develop, and maintain software applications. These professionals use their programming skills to create solutions that meet the needs of businesses and consumers. While this course does not directly teach programming, the mathematical concepts covered, such as logic, algebra, and problem-solving, can provide a strong foundation for those interested in pursuing a career as a Software Engineer.

See salaries and explore the career path for Software Engineer

Computer Scientist

Computer Scientists research and develop new computing technologies and applications. These professionals use their knowledge of mathematics, algorithms, and data structures to design and implement software solutions. While this course does not directly teach computer science, the mathematical concepts covered, such as logic, algebra, and problem-solving, can provide a strong foundation for those interested in pursuing a career as a Computer Scientist.

See salaries and explore the career path for Computer Scientist