MIT FILMED THE LECTURE WHERE THEIR MOST LEGENDARY LINEAR ALGEBRA PROFESSOR EXPLAINS WHY EVERY GPS SIGNAL EVERY MEDICAL SCAN AND EVERY AI MODEL ON EARTH IS SECRETLY SOLVING A PROBLEM THAT HAS NO SOLUTION This is Gilbert Strang, MIT, 18.06 Linear Algebra, lecture 15. The professor whose textbooks sit in over 3,000 universities across 135 countries and whose course has been watched by millions of engineers and mathematicians worldwide. He opens with one claim - when equations have no solution, you do not give up. You solve the closest problem that can be solved. He starts with a vector b and a line through a. The projection of b onto that line is the point closest to b that actually lives on the line. The key is a right angle. The error - the gap between b and its projection - must be perpendicular to a. That one geometric fact produces the entire formula. The projection is a times a transpose b divided by a transpose a. One fraction. The numerator is a matrix. The denominator is a number. Then the projection matrix. Every projection is carried out by a matrix P that multiplies b and lands on the line. That matrix has two properties that define all projections everywhere. First, P transpose equals P - the matrix is symmetric. Second, P squared equals P - project twice and you stay where you are. The first projection moves you onto the line. The second leaves you there. Then higher dimensions. Instead of projecting onto a line, project onto a plane. Instead of one vector a, you have a matrix A with two columns spanning the plane. The error vector must now be perpendicular to both columns - perpendicular to the entire plane. That condition produces one matrix equation. A transpose times the error equals zero. The error lives in the null space of A transpose, which is exactly perpendicular to the column space of A. The four fundamental subspaces appear without being called. Then the formula for n dimensions. The projection matrix is A times A transpose A inverse times A transpose. Strang warns the room not to simplify it by canceling A inverse against A - A is not square, it has no inverse, and the formula cannot be pulled apart. The only case where it simplifies to the identity is when A spans the whole space and the projection is b itself. Watch the moment he applies the formula to fitting a line through three points that no line can pass through. Three equations, two unknowns, no solution. He writes the matrix, writes the right-hand side, and shows that multiplying both sides by A transpose produces an equation that always has a solution. That equation is the foundation of least squares, GPS triangulation, and every regression model trained in 2026. A data scientist I know rewatched this lecture before building a recommendation system from scratch. Said it was the first time the normal equations felt like geometry rather than a formula someone handed down to memorize.