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
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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
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