NE 112 Linear algebra for nanotechnology engineers
Glossary and notation
A course glossary is available here, and a list of notations used in this course is availabe here. It contains many of the new terms introduced in this course. If you find a term missing, please contact the instructor.
Alternative lecture materials
Just in case you don't like my presentation, here are others.
We start with Prof. Jim Hefferon's course which appears in two parts on YouTube with Part 1 and Part 2. More information about the course is available on his website and he also has an online textbook.
You really should look at Stephen Chew's Learning How to Learn series and also good is 3-Blue-and-1-Brown's Linear Algebra series.
Lecture materials
The course book is available as a free Adobe portable document format (pdf):
An introduction to linear algebra for nanotechnology engineering with applications in Matlab
Note that this pdf uses bookmarks, so you are able to navigate to specific chapters, sections and subsections directly from the pdf viewer. Additionally, you can select an item in the table of contents and the pdf viewer will take you to that point in the on-line text.
The course is broken into twelve weeks, and unfortunately, some week breaks do interrupt a particular topic; however, if you are learning this material on your own, this really should not pose an issue. At the end of each week, an assignment is posted without solutions.
| Topic | Video | Time | PDFs | Week | Questions, sample code, Wikipedia, and other links |
|
|---|---|---|---|---|---|---|
| 0 | Course introduction | YouTube | (15 min) | Slides PPTX |
1 | |
Chapter 1
| Topic | Video | Time | PDFs | Week | Questions, sample code, Wikipedia, and other links |
|
|---|---|---|---|---|---|---|
| 1.1 | Multiplication tables | YouTube | (3 min) | Slides PPTX |
1 | |
| 1.2 | Greek letters | YouTube | (3 min) | Slides PPTX |
1 | |
| 1.3 | Sequences, sums and series | YouTube | (33 min) | Slides PPTX |
1 | |
| 1.4 | Matlab (optional) | 1 | ||||
| 1.5 | A brief introduction to the axiomatic method | YouTube | (10 min) | Slides PPTX |
1 | |
Section 1.6 Fields and complex numbers | ||||||
| 1.6 | Introduction to fields and complex numbers | 1 | ||||
| 1.6.1 | The field axioms | YouTube | (21 min) | Slides PPTX |
1 | |
| 1.6.2 | Introducing j | YouTube | (18 min) | Slides PPTX |
1 | |
| 1.6.2a | i versus j (optional) - math versus engineering |
YouTube | (8 min) | Slides PPTX |
1 | |
| 1.6.3 | Complex numbers and their components | YouTube | (9 min) | Slides PPTX |
1 | |
| 1.6.4 | Geometric interpretation of complex numbers | YouTube | (9 min) | Slides PPTX |
1 | |
| 1.6.5 | The absolute value of a complex number | YouTube | (13 min) | Slides PPTX |
1 | |
| 1.6.6 | The angle or argument of a complex number and the polar representation | YouTube | (10 min) | n/a | 1 | |
| 1.6.7 | Switching between rectangular and polar representations | YouTube | (7 min) | Slides PPTX |
2 | |
Sub-section 1.6.8 Complex arithmetic | ||||||
| 1.6.8 | Introduction to complex arithmetic | |||||
| 1.6.8.1 | Addition of complex numbers | YouTube | (7 min) | Slides PPTX |
2 | |
| 1.6.8.2 | Additive inverse of complex numbers | YouTube | (5 min) | Slides PPTX |
2 | |
| 1.6.8.3 | Subtraction of complex numbers | YouTube | (8 min) | Slides PPTX |
2 | |
| 1.6.8.4 | Multiplication of complex numbers | YouTube | (19 min) | Slides PPTX |
2 | |
| 1.6.8.5 | The complex conjugate | YouTube | (14 min) | Slides PPTX |
2 | |
| 1.6.8.6 | The multiplicative inverse of complex numbers | YouTube | (18 min) | Slides PPTX |
2 | |
| 1.6.8.7 | Division of complex numbers | YouTube | (10 min) | Slides PPTX |
2 | |
| 1.6.8.8 | Integer powers of complex numbers | YouTube | (18 min) | Slides PPTX |
2 | |
| 1.6.8.9 | The square root of complex numbers | YouTube | (7 min) | Slides PPTX |
2 | |
| 1.6.9 | The complex numbers are a field | YouTube | (7 min) | Slides PPTX |
2 | |
| 1.6.10 | Three inequalities | YouTube | (18 min) | Slides PPTX |
2 | |
| 1.6.11 | The fundamental theorem of algebra | YouTube | (13 min) | Slides PPTX |
2 | |
| 1.6.12 | Roots of unity (optional) | YouTube | (11 min) | Slides PPTX |
2 | |
| 1.6.13 | Roots of polynomials with real coefficients | YouTube | (14 min) | Slides PPTX |
2 | |
| 1.6.14 | Geometric series | YouTube | (6 min) | Slides PPTX |
3 | |
| 1.6.15 | The exponential and trigonmetric functions (optional) | YouTube | (21 min) | Slides PPTX |
3 | |
| 1.6.16 | Fields in this course | - | 3 | |||
| 1.6.17 | An application to electrical engineering (optional) | - | 3 | |||
| 1.6.18 | Complex numbers in this course | 3 | ||||
| 1.6.19 | Application of complex numbers | 3 | ||||
| 1.6.21 | Summary of fields | 3 | ||||
| 1.7 | Summary of introductory material | 3 | ||||
Chapter 2: Vectors and Vector spaces
| Topic | Video | Time | PDFs | Week | Questions, sample code, Wikipedia, and other links |
|
|---|---|---|---|---|---|---|
Section 2.1 Real n-dimensional vectors | ||||||
| 2.1 | Real n-dimensional vectors | 3 | ||||
| 2.1.1 | Definition of real n-dimensional vectors | YouTube | (9 min) | Slides PPTX |
3 | |
| 2.1.2 | The zero vector | YouTube | (3 min) | Slides PPTX |
3 | |
| 2.1.3 | Geometric interpretaion of real 2- and 3-dimensional vectors | YouTube | (6 min) | Slides PPTX |
3 | |
| 2.1.4 | Geometric interpretaion of real higher-dimensional vectors | YouTube | (6 min) | Slides PPTX |
3 | |
| 2.1.5 | Equality of vectors | YouTube | (6 min) | Slides PPTX |
3 | |
| 2.1.7 | Two-dimensional vectors versus complex numbers | YouTube | (8 min) | Slides PPTX |
3 | |
| 2.2 | Complex n-dimesnional vectors | YouTube | (4 min) | Slides PPTX |
3 | |
Section 2.3 Vector operations | ||||||
| 2.3 | Vector operators | YouTube | (3 min) | Slides PPTX |
3 | |
| 2.3.1 | Scalar multiplication | YouTube | (10 min) | Slides PPTX |
3 | |
| 2.3.2 | Vector addition | YouTube | (9 min) | Slides PPTX |
3 | |
| 2.3.3 | Properties and axioms of vector spaces | YouTube | (40 min) | Slides PPTX |
4 | |
| 2.3.4 | Linear combinations of vectors | YouTube | (7 min) | Slides PPTX |
4 | |
| 2.4 | Other vector spaces | YouTube | (23 min) | Slides PPTX |
4 | |
Chapter 3: Subspaces
| Topic | Video | Time | PDFs | Week | Questions, sample code, Wikipedia, and other links |
|
|---|---|---|---|---|---|---|
| 3.1 | A review of sets | YouTube | (15 min) | Slides PPTX |
4 | |
| 3.2 | When is a subset also a subspace | YouTube | (31 min) | Slides PPTX |
4 | |
| 3.3 | Examples of subspaces | YouTube | (20 min) | Slides PPTX |
4 |
Chapter 4: Normed vector spaces
| Topic | Video | Time | PDFs | Week | Questions, sample code, Wikipedia, and other links |
|
|---|---|---|---|---|---|---|
| 4 | Introduction to normed vector spaces | YouTube | (2 min) | Slides PPTX |
4 | |
| 4.1 | The 2-norm for finite-dimensional vectors | YouTube | (12 min) | Slides PPTX |
4 | |
| 4.2 | Other norms for finite-dimensional vectors | YouTube | (8 min) | Slides PPTX |
4 | |
| 4.3 | Unit vectors and normalization | YouTube | (7 min) | Slides PPTX |
4 | |
| 4.4 | Norms in other vector spaces | YouTube | (9 min) | Slides PPTX |
4 |
Chapter 5: Inner product spaces
| Topic | Video | Time | PDFs | Week | Questions, sample code, Wikipedia, and other links |
|
|---|---|---|---|---|---|---|
| 5 | Introduction to inner product spaces | YouTube | (4 min) | Slides PPTX |
4 | |
| 5.1 | The inner product for finite-dimensional vectors | YouTube | (13 min) | Slides PPTX |
4 |
|
| 5.2 | The norm induced by the inner product | YouTube | (3 min) | Slides PPTX |
4 | |
| 5.3 | Inner products on other vector spaces | YouTube | (8 min) | Slides PPTX |
5 | |
| 5.4 | Orthogonality | YouTube | (11 min) | Slides PPTX |
5 | |
| 5.5 | Orthogonality in other vector spaces (optional) | 5 | ||||
| 5.6 | The Pythagorean theorem | YouTube | (11 min) | Slides PPTX |
5 | |
| 5.7 | Projections and best approximations | YouTube | (45 min) | Slides PPTX |
5 |
|
| 5.8 | The Cauchy-Bunyakovsky-Schwarz inequality | YouTube | (13 min) | Slides PPTX |
5 | |
| 5.9 | The angle between two vectors | YouTube | (10 min) | Slides PPTX |
5 | |
| 5.10 | The Gram-Schmidt algorithm | YouTube | (28 min) | Slides PPTX |
5 |
|
Chapter 6: Solutions to systems of linear equations, linear independence and bases
| Topic | Video | Time | PDFs | Week | Questions, sample code, Wikipedia, and other links |
|
|---|---|---|---|---|---|---|
| 6.1 | Equations, linear equations and systems of equations | YouTube | (21 min) | Slides PPTX |
5 | |
| 6.2 | Finding solutions to systems of linear equations | YouTube | (24 min) | Slides PPTX |
5 | |
| 6.3 | Linear combinations of vectors | YouTube | (21 min) | Slides PPTX |
5 | |
| 6.4 | A systematic approach to solving a system of linear equations | YouTube | (3 min) | Slides PPTX |
6 | |
| 6.4.1 | Matrices | YouTube | (10 min) | Slides PPTX |
6 | |
| 6.4.2 | Classification of matrices | YouTube | (6 min) | Slides PPTX |
6 | |
| 6.4.3 | Augmented matrices | YouTube | (8 min) | Slides PPTX |
6 | |
| 6.4.4 | Zero rows, leading zeros and leading non-zero entries | YouTube | (6 min) | Slides PPTX |
6 | |
| 6.4.5 | Row-echelon form | YouTube | (12 min) | Slides PPTX |
6 | |
| 6.4.6 | Solutions to systems of linear equations with backward substitution | YouTube | (53 min) | Slides PPTX |
6 | |
| 6.4.7 | Row operations | YouTube | (32 min) | Slides PPTX |
7 | |
| 6.4.8 | Gaussian elimination with partial pivoting | YouTube | (67 min) | Slides PPTX |
7 |
|
| 6.4.9 | Solutions to homogeneous systems of linear equations | YouTube | (29 min) | Slides PPTX |
7 | |
| 6.5 | Linear dependence and independence of vectors | YouTube | (36 min) | Slides PPTX |
7 | |
| 6.6 | Spans and subspaces | YouTube | (20 min) | Slides PPTX |
8 | |
| 6.7 | Bases and dimension | YouTube | (26 min) | Slides PPTX |
8 | |
| 6.8 | Vectors as coefficients of a basis (optional) | YouTube | - | Slides PPTX |
8 |
Chapter 8: Linear maps and linear operators
| Topic | Video | Time | PDFs | Week | Questions, sample code, Wikipedia, and other links |
|
|---|---|---|---|---|---|---|
| 8 | Introduction to linear maps and linear operators | YouTube | (11 min) | Slides PPTX |
8 | |
| 8.1 | The superposition principle | YouTube | (8 min) | Slides PPTX |
8 | |
| 8.2 | Definition of linear maps and linear operators | YouTube | (35 min) | Slides PPTX |
8 | |
| 8.4 | Special linear maps and linear operators | YouTube | (10 min) | Slides PPTX |
8 | |
| 8.5 | The range of a linear map | YouTube | (17 min) | Slides PPTX |
8 | |
| 8.6 | The null space of a linear map | YouTube | (32 min) | Slides PPTX |
8 | |
| 8.8 | The vector space of linear maps | YouTube | (35 min) | Slides PPTX |
8 |
|
| 8.9 | The composition of linear maps | YouTube | (27 min) | Slides PPTX |
9 |
|
| 8.10 | Linear operators | YouTube | (22 min) | Slides PPTX |
9 |
|
| 8.11 | Row operations | YouTube | (17 min) | Slides PPTX |
9 |
|
Chapter 9: Linear operators on finite-dimensional vector spaces
| Topic | Video | Time | PDFs | Week | Questions, sample code, Wikipedia, and other links |
|
|---|---|---|---|---|---|---|
| 9 | Introduction to linear operators on finite-dimensional vector spaces | YouTube | (4 min) | Slides PPTX |
9 | |
| 9.1 | The determinant | YouTube | (38 min) | Slides PPTX |
9 |
|
| 9.2 | The trace | YouTube | (8 min) | Slides PPTX |
9 |
|
| 9.3 | The inverse | YouTube | (28 min) | Slides PPTX |
9 |
|
| 9.4 | Matrices of orthonormal vectors and their inverses | YouTube | (22 min) | Slides PPTX |
9 |
Chapter 10: Matrix decompositions
| Topic | Video | Time | PDFs | Week | Questions, sample code, Wikipedia, and other links |
|
|---|---|---|---|---|---|---|
| 10 | Introduction to matrix decompositions | YouTube | (14 min) | Slides PPTX |
10 |
|
| 10.1 | PLU decompositions | YouTube | (37 min) | Slides PPTX |
10 |
Chapter 11: The adjoint of linear maps
| Topic | Video | Time | PDFs | Week | Questions, sample code, Wikipedia, and other links |
|
|---|---|---|---|---|---|---|
| 11 | Introduction to the adjoint of a linear map | YouTube | (3 min) | Slides PPTX |
10 | |
| 11.1 | The properties of the adjoint | YouTube | (18 min) | Slides PPTX |
10 | |
| 11.2 | The adjoint of a linear map on real finite-dimensional vector spaces (real matrices) | YouTube | (23 min) | Slides PPTX |
10 |
|
| 11.3 | The adjoint of a linear map on complex finite-dimensional vector spaces (complex matrices) | YouTube | (10 min) | Slides PPTX |
10 | |
| 11.4 | An alternative interpretation of the inner product | YouTube | (5 min) | Slides PPTX |
10 | |
| 11.5 | Self-adjoint and skew-adjoint linear operators | YouTube | (5 min) | Slides PPTX |
10 |
|
| 11.7 | Normal operators | YouTube | (17 min) | Slides PPTX |
10 | |
| 11.8 | Isometric operators | YouTube | (24 min) | Slides PPTX |
10 |
|
| 11.9 | Linear regression and least squares | YouTube | (25 min) | Slides PPTX |
10 |
Chapter 12: Eigenvalues and eigenvectors
| Topic | Video | Time | PDFs | Week | Questions, sample code, Wikipedia, and other links |
|
|---|---|---|---|---|---|---|
| 12.1 | Invariant subspaces, eigenvalues and eigenvectors | YouTube | (50 min) | Slides PPTX |
11 | |
| 12.2 | Eigenvalues and eigenvectors of operators on signals | YouTube | (26 min) | Slides PPTX |
11 |
|
| 12.3 | Eigenvalue decomposition or diagonalization | YouTube | (26 min) | Slides PPTX |
11 |
|
| 12.4 | Positive semi-definite matrices and finding the maximum eigenvalue | YouTube | (27 min) | Slides PPTX |
11 | |
| 12.5 | Powers of matrices | YouTube | (17 min) | Slides PPTX |
11 | |
| 12.6 | Algorithms for finding the eigenvalues of a matrix (optional) | YouTube | (18 min) | Slides PPTX |
11 | |
| 12.7 | Solving systems of homogeneous linear ordinary differential equations (optional) | YouTube | (26 min) | Slides PPTX |
11 | |
| 12.8 | Principal component analysis (optional) | YouTube | (21 min) | Slides PPTX |
11 |
Chapter 7: A digression to real 3-dimensional space
| Topic | Video | Time | PDFs | Week | Questions, sample code, Wikipedia, and other links |
|
|---|---|---|---|---|---|---|
| 7 | Introduction to this digression to real 3-dimensional space | YouTube | (4 min) | Slides PPTX |
12 | |
| 7.1 | The vector representation of a line | YouTube | (9 min) | Slides PPTX |
||
| 7.2 | Finding the line through two points | YouTube | (4 min) | Slides PPTX |
12 | |
| 7.3 | The vector representation of a plane | YouTube | (11 min) | Slides PPTX |
12 | |
| 7.4 | The equation of a plane | YouTube | (14 min) | Slides PPTX |
12 | |
| 7.5 | The cross product | YouTube | (26 min) | Slides PPTX |
12 | |
| 7.6 | The equatios of a line: the intersection of two planes | YouTube | (15 min) | Slides PPTX |
12 | |
| 7.7 | The plane containing three points | YouTube | (6 min) | Slides PPTX |
12 |
In-person mid-term examination formula sheet
A course summary is available here:
This is not guaranteed to cover everything that may be covered on the final, but this summarizes many of the definitions, algorithms and theorems in the course.
Here are two files that describe finding the properties of linear mappings, including the images, range, null space, and pre-images:
Previous final examinations:
A practice mid-term from last year and solutions are available here:
- 2016 practice midterm
- 2016 solutions
- 2017 practice midterm
- 2017 solutions
- 2018 practice midterm
- 2018 solutions
The answer to 5 should be 135 - 923j.
Previous mid-terms:
A primer on vectors
The document A primer on vectors is an introduction to vectors for those students whose school boards do not teach vectors in secondary school. It introduces vectors to the extent necessary to get up to speed in this course.
2. Introduction to two-dimensional vectors
- 2.1 Two-dimensional vectors
- 2.2 The norm of a vector
- 2.3 Unit vectors
- 2.4 Vector equality
- 2.5 Vector addition
- 2.6 Scalar multiplication
- 2.7 Vector difference and distance
- 2.8 Normalizing a vector
- 2.9 Orthogonal vectors
- 2.10 The inner product
- 2.11 The inner product and the 2-norm
- 2.12 The inner product and orthogonality
- 2.13 Vectors orthogonal to a given vector
For your additional reading
- A primer on vectors
- Trigonometry made somewhat easier... and an associated video on YouTube
- An easier way of remembering various trigonometric values
- How large is a coulomb?
NE 112
Department of Electrical and Computer Engineering
University of Waterloo
200 University Avenue West
Waterloo, Ontario, Canada N2L 3G1
Department of Electrical and Computer Engineering
University of Waterloo
200 University Avenue West
Waterloo, Ontario, Canada N2L 3G1