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Least Squares Solutions and Deriving the Normal Equation | Linear Algebra

We introduce the least squares problem and how to solve it using the techniques of linear algebra. We'll discuss least squares error and the least squares error vector. The idea is that we are finding the best approximation, in a subspace W, of a vector u that does not lie in the subspace. We'll find that the vector we seek is the projection of u onto the subspace W. This is the best approximation theorem, which we prove. We'll see how it applies in R^n to solve the least squares problem for systems of linear equations, and find it leads to something called the normal equation for a matrix. By using the normal equation, we'll find the least squares solution for two inconsistent systems Ax=b. Our first example has a unique least squares solution and our second example has infinitely many least squares solutions, which we will represent parametrically. #linearalgebra Pythagorean Theorem in Inner Product Spaces:    • Generalized Pythagorean Theorem in In...   Linear Algebra course:    • Linear Algebra   Linear Algebra exercises:    • Linear Algebra Exercises   Join Wrath of Math to get exclusive videos, music, and more:    / @wrathofmath   0:00 Intro 0:53 An Inconsistent System and Why to Solve It 2:43 Least Squares Solutions and Least Squares Error 3:57 Why is it "Least Squares"? 5:27 Seeing the Solution 9:28 Best Approximation Theorem in Inner Product Spaces 13:21 Best Approximation Theorem in R^n 14:34 Deriving the Normal Equation 17:00 Consistency of the Normal Equation 18:21 Full Least Squares Example (Unique Solution) 21:06 Full Least Squares Example (Infinitely Many Solutions) 23:40 Conclusion ◉Textbooks I Like◉ Graph Theory: https://amzn.to/3JHQtZj Real Analysis: https://amzn.to/3CMdgjI Abstract Algebra: https://amzn.to/3IjoZaO Linear Algebra: https://amzn.to/43xAWEz Calculus: https://amzn.to/3PieD1M Proofs and Set Theory: https://amzn.to/367VBXP (available for free online) Statistics: https://amzn.to/3tsaEER Discrete Math: https://amzn.to/3qfhoUn Number Theory: https://amzn.to/3JqpOQd ★DONATE★ ◆ Support Wrath of Math on Patreon for early access to new videos and other exclusive benefits:   / wrathofmathlessons   ◆ Donate on PayPal: https://www.paypal.me/wrathofmath Thanks to Loke Tan, Raül Beienheimer, Matt Venia, Micheline, Doug Walker, Odd Hultberg, Marc, Shlome Ashkenazi, Barbora Sharrock, Mohamad Nossier, Rolf Waefler, Shadow Master, and James Mead for their generous support on Patreon! Outro music is mine. You cannot find it anywhere, for now. Follow Wrath of Math on... ● Instagram:   / wrathofmathedu   ● Facebook:   / wrathofmath   ● Twitter:   / wrathofmathedu   My Math Rap channel:    / @mathbars2020