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6: Orthogonality and Least Squares
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6.1: The Dot Product
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6.2: Orthogonal Complements and the Matrix Tranpose
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This section introduces the notion of an orthogonal complement, the set of vectors each of which is orthogonal to a prescribed subspace. We'll also find a way to describe dot products using matrix products, which allows us to study orthogonality using many of the tools for understanding linear systems that we developed earlier.
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6.3: Orthogonal bases and projections
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6.4: Finding Orthogonal Bases
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6.5: Orthogonal Least Squares
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n this section, we'll explore how the techniques developed in this chapter enable us to find the line that best approximates the data. More specifically, we'll see how the search for a line passing through the data points leads to an inconsistent system Ax=b. Since we are unable to find a solution x, we instead seek the vector x where Ax is as close as possible to b. Orthogonal projection give us just the right tool for doing this.