Gilbert Strang and Linear Algebra made easy
If you don't know what linear algebra is, then you must have had the same kind of math teacher I had. Linear algebra is the part of math that deals with matrices. They are extremely useful in many area of computer science: computer graphics, simulation and machine learning. They're a basic tool that one must master in order to understand what one is doing in those fields.
A shearing applied to La Gioconda.
Yet, I noticed that it remains a poorly understood field of mathematics, even by seasoned engineer, which is a shame. Indeed, math teachers tend to teach that subject in a very dry way. They'll tend to focus on the resolution of system of equations and spending a lot of time on definition and formal proof, which end up confusing the listener. I guess it depends in which country you went to school to, but I had to learn by heart how to solve a limited set of typical problems that would then have to be solved again without my textbook during the exam. That is really not the best way to understand something. It's just a memory exercise.
Enter Gilbert Strang. I stumbled upon his website a long time ago and started watching his LinAlg lectures. Gilbert Strang is a Professor of Mathematics at the Massachusetts Institute of Technology (MIT). He's been teaching there for decades and have a specialty in LinAlg. He honed a well rehearsed LinAgl class for years and is one of the first teacher to have published his lectures online at the end of the 90s.
Prof. Strang's lectures are remarkable on many levels. First, he's not focused on demonstration or definition, even though they are effectively present in his lectures. It happens that they are not at the center of its teaching but rather a consequence of your own understanding. Prof. Strang will insist on the many interpretation of a fact, giving you the various aspect of a mathematical truth. He will show you the dots and then help you connect them. This is clearly visible in the first video of the course which is entitled the "Geometry of Linear Algebra". Right into it, Prof. Strang will give you the geometric interpretation of a matrix. What the rows represent, what the columns represent. He's not jumping on the system of equation bandwagon but only to tell you that's not what this is about. What this is about is manipulating objects which, although conceptual, have a real meaning in an every day engineer's life.
The fundamental truth that you will quickly grasp, thanks to Prof. Strang clear enunciation, is that matrices are a way to transform vectors. Be it rotation, skewing and even translation in case of homogeneous matrices, what you are doing when using a matrix is move things. Once you understand this fundamental and clearly simple concept then everything makes sense. A matrix is about changing perspective or, mathematically, changing your coordinate system. A matrix inverse is about putting an object back where it was. Finding the rank is understanding how you matrix preserve or transform the spaces in which your objects live in. Computing an eigenvector is extracting, from this seemingly formless bag of numbers, the axe upon which the transformation revolves around. Even the somehow complex notion of Singular Value Decomposition becomes clear and intuitive.
Prof. Strang strength relies on many factors but one important aspect is his empathy. You can clearly see that he thinks with the mind of a student. Even though the technique of lecture ex cathedra is not particularly innovative, his diction, his posture, his flow of thinking are all guided toward the goal of taking your hand and slowly but surely remove you from the darkness of confusion to the warming light of understanding. Many times during the course I caught myself smiling, even laughing sometimes at the discovery of a seemingly simple truth which had eluded me all these years.
One important aspect of Prof. Strang's class is that it comes with its own textbook. The class matches the chapter and the book is definitely a necessary companion to the lectures. It goes deeper on certain concept, clarifies some points and provide exercises which, let's be blunt, are necessary if you ever want to master and remember what you are learning.
Introduction to Linear Algebra, Fifth Edition (Gilbert Strang) Fifth Edition
Now spending time watching Prof. Strang's lectures while reading the book is certainly a substantial investment of your time. But an investment which will be well rewarded in the clearer understanding of one of the most fundamental tool of engineering.