Linear Algebra

Linear algebra is taught almost everywhere as a set of procedures. Multiply these numbers in this order, take this determinant, row-reduce until it stops. People pass the exam and come away able to compute things they cannot picture, which is why so many capable students describe the subject as arbitrary. It is not arbitrary. Almost every operation in it is a simple geometric idea wearing a numerical disguise.

This course is Grant Sanderson's Essence of Linear Algebra, in order and complete. It is unusually well suited to being a course rather than a playlist, because each chapter genuinely depends on the one before it: a linear transformation is defined in chapter three, matrix multiplication is revealed as composing two of them in chapter four, and the determinant in chapter six is meaningless without both. Watching these out of order is most of the reason people bounce off the series.

Be clear about what it is for. This teaches you to see what the operations mean; it does not drill you on performing them. That is a deliberate and correct division of labour — you can get computational practice from any textbook or problem set, and almost nowhere else will you find the intuition. Used alongside a conventional course it is transformative. Used instead of one, you will understand a great deal and still need practice before an exam.

About three hours in the core sequence, in ten-minute pieces, with another twenty-three available in the optional modules that follow it — two full college courses, the computational side that 3Blue1Brown deliberately leaves out, and four of Gilbert Strang's MIT lectures. The core sequence repays being watched slowly. Pause when the animation moves and predict what it will do before it does. Chapters nine through eleven on dot and cross products are the ones to take at half speed; chapter fourteen on eigenvectors is where everything before it pays off at once.