## About this Course

288,856 recent viewsIn this course on Linear Algebra we look at what linear algebra is and how it relates to vectors and matrices. Then we look through what vectors and matrices are and how to work with them, including the knotty problem of eigenvalues and eigenvectors, and how to use these to solve problems. Finally we look at how to use these to do fun things with datasets - like how to rotate images of faces and how to extract eigenvectors to look at how the Pagerank algorithm works.

#### Flexible deadlines

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#### Shareable Certificate

Earn a Certificate upon completion

#### 100% online

Start instantly and learn at your own schedule.

#### Course 1 of 3 in the

**Mathematics for Machine Learning Specialization**

#### English

Subtitles: Arabic, French, Portuguese (European), Italian, Vietnamese, German, Russian, English, Spanish

Skills you will gain

- Eigenvalues And Eigenvectors
- Basis (Linear Algebra)
- Transformation Matrix
- Linear Algebra

#### Flexible deadlines

Reset deadlines in accordance to your schedule.

#### Shareable Certificate

Earn a Certificate upon completion

#### 100% online

Start instantly and learn at your own schedule.

#### Course 1 of 3 in the

**Mathematics for Machine Learning Specialization**

#### English

Subtitles: Arabic, French, Portuguese (European), Italian, Vietnamese, German, Russian, English, Spanish

## Instructors

### David Dye

Professor of MetallurgyDepartment of Materials**302,202**Learners**2**Courses

### Samuel J. Cooper

Associate ProfessorDyson School of Design Engineering**302,202**Learners**2**Courses

### A. Freddie Page

Strategic Teaching FellowDyson School of Design Engineering**302,202**Learners**2**Courses## Offered by

### Imperial College London

Imperial College London is a world top ten university with an international reputation for excellence in science, engineering, medicine and business. located in the heart of London. Imperial is a multidisciplinary space for education, research, translation and commercialisation, harnessing science and innovation to tackle global challenges.

## Week 1

2 hours to complete

## Introduction to Linear Algebra and to Mathematics for Machine Learning

In this first module we look at how linear algebra is relevant to machine learning and data science. Then we'll wind up the module with an initial introduction to vectors. Throughout, we're focussing on developing your mathematical intuition, not of crunching through algebra or doing long pen-and-paper examples. For many of these operations, there are callable functions in Python that can do the adding up - the point is to appreciate what they do and how they work so that, when things go wrong or there are special cases, you can understand why and what to do.

2 hours to complete

5 videos (Total 28 min), 4 readings, 3 quizzes

**5 videos**

**Introduction: Solving data science challenges with mathematics**2m

Motivations for linear algebra3m

Getting a handle on vectors9m

Operations with vectors11m

Summary1m

**4 readings**

About Imperial College & the team5m

How to be successful in this course5m

Grading policy5m

Additional readings & helpful references10m

**3 practice exercises**

Exploring parameter space20m

Solving some simultaneous equations15m

Doing some vector operations30m

## Week 2

2 hours to complete

## Vectors are objects that move around space

In this module, we look at operations we can do with vectors - finding the modulus (size), angle between vectors (dot or inner product) and projections of one vector onto another. We can then examine how the entries describing a vector will depend on what vectors we use to define the axes - the basis. That will then let us determine whether a proposed set of basis vectors are what's called 'linearly independent.' This will complete our examination of vectors, allowing us to move on to matrices in module 3 and then start to solve linear algebra problems.

2 hours to complete

8 videos (Total 44 min)

**8 videos**

**Introduction to module 2 - Vectors**49s

Modulus & inner product10m

Cosine & dot product5m

Projection6m

Changing basis11m

Basis, vector space, and linear independence4m

Applications of changing basis3m

Summary1m

**4 practice exercises**

Dot product of vectors15m

Changing basis15m

Linear dependency of a set of vectors15m

Vector operations assessment15m

## Week 3

3 hours to complete

## Matrices in Linear Algebra: Objects that operate on Vectors

Now that we've looked at vectors, we can turn to matrices. First we look at how to use matrices as tools to solve linear algebra problems, and as objects that transform vectors. Then we look at how to solve systems of linear equations using matrices, which will then take us on to look at inverse matrices and determinants, and to think about what the determinant really is, intuitively speaking. Finally, we'll look at cases of special matrices that mean that the determinant is zero or where the matrix isn't invertible - cases where algorithms that need to invert a matrix will fail.

3 hours to complete

8 videos (Total 57 min)

**8 videos**

**Matrices, vectors, and solving simultaneous equation problems**5m

How matrices transform space5m

Types of matrix transformation8m

Composition or combination of matrix transformations8m

Solving the apples and bananas problem: Gaussian elimination8m

Going from Gaussian elimination to finding the inverse matrix8m

Determinants and inverses10m

Summary59s

**2 practice exercises**

Using matrices to make transformations30m

Solving linear equations using the inverse matrix30m

## Week 4

7 hours to complete

## Matrices make linear mappings

In Module 4, we continue our discussion of matrices; first we think about how to code up matrix multiplication and matrix operations using the Einstein Summation Convention, which is a widely used notation in more advanced linear algebra courses. Then, we look at how matrices can transform a description of a vector from one basis (set of axes) to another. This will allow us to, for example, figure out how to apply a reflection to an image and manipulate images. We'll also look at how to construct a convenient basis vector set in order to do such transformations. Then, we'll write some code to do these transformations and apply this work computationally.

7 hours to complete

6 videos (Total 53 min)

**6 videos**

**Introduction: Einstein summation convention and the symmetry of the dot product**9m

Matrices changing basis11m

Doing a transformation in a changed basis4m

Orthogonal matrices6m

The Gram–Schmidt process6m

Example: Reflecting in a plane14m

**2 practice exercises**

Non-square matrix multiplication20m

Example: Using non-square matrices to do a projection30m

## Week 5

4 hours to complete

## Eigenvalues and Eigenvectors: Application to Data Problems

Eigenvectors are particular vectors that are unrotated by a transformation matrix, and eigenvalues are the amount by which the eigenvectors are stretched. These special 'eigen-things' are very useful in linear algebra and will let us examine Google's famous PageRank algorithm for presenting web search results. Then we'll apply this in code, which will wrap up the course.

4 hours to complete

9 videos (Total 44 min), 1 reading, 5 quizzes

**9 videos**

**Welcome to module 5**52s

What are eigenvalues and eigenvectors?4m

Special eigen-cases3m

Calculating eigenvectors10m

Changing to the eigenbasis5m

Eigenbasis example7m

Introduction to PageRank8m

Summary1m

Wrap up of this linear algebra course1m

**1 reading**

Did you like the course? Let us know!10m

**4 practice exercises**

Selecting eigenvectors by inspection20m

Characteristic polynomials, eigenvalues and eigenvectors30m

Diagonalisation and applications20m

Eigenvalues and eigenvectors25m