So, I'm cruising through Linear Algebra when I get to Orthogonality. You'd think this would be the easiest chapter in the book, since I do least squares stuff all the time at work.
You'd be wrong.
Problem is, I'm so familiar with how least squares projections work that my brain has completely shelved the theory supporting it. So, fine, I can find a least squares fit in Rn. Now define the inner product space of all possible solutions. Find the null space. Show that integrating the product of functions f, g from a to b forms a valid inner product on the space of functions continuous in (a,b). Find the subordinate matrix norm. Compute an orthonormal set using that inner product that spans the basic trig functions on (0, π/4). You get the idea. Sure, I knew all this stuff 30 years ago. Haven't done much but project polynomials ever since.
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