Explore the concept of least squares solutions and the derivation of the normal equation in this 25-minute linear algebra video. Learn about least squares error, the best approximation theorem, and how to solve inconsistent systems of linear equations. Discover the application of the Pythagorean theorem in inner product spaces and its relevance to finding optimal solutions. Follow along with detailed examples, including cases with unique and infinitely many solutions. Gain a deeper understanding of linear algebra concepts through clear explanations and visual demonstrations. Perfect for students and enthusiasts looking to enhance their knowledge of advanced mathematical techniques.