Linear algebra for people building AI, taught with pictures, plain words, and code.https://linearly.space/https://linearly.space/lectures/lecture-00/https://linearly.space/lectures/lecture-00/Computing spent decades following exact recipes. Learning began when we replaced the recipe with a search, and the search runs on arrays of numbers.https://linearly.space/lectures/lecture-01/https://linearly.space/lectures/lecture-01/What a dimension is, how vectors combine, and why the equation Ax = b runs through the whole course.https://linearly.space/lectures/lecture-02/https://linearly.space/lectures/lecture-02/A matrix is a machine that moves every point of space at once. Its columns say everything about it, and some machines destroy information.https://linearly.space/lectures/lecture-03/https://linearly.space/lectures/lecture-03/Every array carries a list of axes. Learning to read that list, and to say which axis should disappear, is most of what writing numerical code is.https://linearly.space/lectures/lecture-04/https://linearly.space/lectures/lecture-04/The algorithm inside every solver. Turn a system into a staircase, read the answer from the bottom up, and discover that every step you took was a matrix.https://linearly.space/lectures/lecture-05/https://linearly.space/lectures/lecture-05/Four readings of AB on one small example, why order matters and grouping does not, and what it takes to undo a matrix at all.https://linearly.space/lectures/lecture-06/https://linearly.space/lectures/lecture-06/Split a matrix into simpler matrices and it starts talking: how many independent columns it really has, and exactly what elimination did to it.https://linearly.space/lectures/lecture-07/https://linearly.space/lectures/lecture-07/Adding and scaling are the only two moves. Any place where they behave is a vector space, and the columns of a matrix build one of the most useful ones.https://linearly.space/lectures/lecture-07-5/https://linearly.space/lectures/lecture-07-5/Add two solutions of Ax = 0 and you get another solution. Add two solutions of Ax = b and you do not. That gap is what makes one of them a subspace, and elimination hands you a basis for it.https://linearly.space/lectures/lecture-08/https://linearly.space/lectures/lecture-08/One anchor plus the whole null space. Whether b fits at all, and how a single number, the rank, tells you how many answers to expect before you compute one.https://linearly.space/lectures/lecture-09/https://linearly.space/lectures/lecture-09/One small matrix, four subspaces, four bases read straight off a single elimination, and the reason row rank always equals column rank.https://linearly.space/lectures/lecture-10/https://linearly.space/lectures/lecture-10/Row space and null space meet at a right angle, and so does the other pair. Nothing perpendicular is missing, every reachable b comes from exactly one x, and that is where the pseudoinverse comes from.https://linearly.space/lectures/lecture-11/https://linearly.space/lectures/lecture-11/When the target is out of reach, the best you can do is its shadow. The projection formula, the projection matrix, and why the miss is always perpendicular.https://linearly.space/lectures/lecture-12/https://linearly.space/lectures/lecture-12/No line goes through all the data. Least squares picks the one that misses by the least, and it is the projection from the last chapter wearing different clothes.https://linearly.space/lectures/lecture-13/https://linearly.space/lectures/lecture-13/Perpendicular unit columns make every formula in this course cheaper. How to build them from any basis, and the factorization that records the work.https://linearly.space/lectures/lecture-14/https://linearly.space/lectures/lecture-14/One number says how much a matrix inflates or flattens space. Every rule, every formula, and every use of the determinant comes out of that.https://linearly.space/lectures/lecture-15/https://linearly.space/lectures/lecture-15/Almost every direction gets turned when a matrix acts. A few survive and only stretch. Those directions run the long run, and their stretch factors shape every loss surface you will ever descend.