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Subsection 4.1.2 Overview

  • 4.1 Opening

    • 4.1.1 Fitting the best line

    • 4.1.2 Overview

    • 4.1.3 What you will learn

  • 4.2 Solution via the Method of Normal Equations

    • 4.2.1 The four fundamental spaces of a matrix

    • 4.2.2 The Method of Normal Equations

    • 4.2.3 Solving the normal equations

    • 4.2.4 Conditioning of the linear least squares problem

    • 4.2.5 Why using the Method of Normal Equations could be bad

  • 4.3 Solution via the SVD

    • 4.3.1 The SVD and the four fundamental spaces

    • 4.3.2 Case 1: \(A \) has linearly independent columns

    • 4.3.3 Case 2: General case

  • 4.4 Solution via the QR factorization

    • 4.4.1 \(A \) has linearly independent columns

    • 4.4.2 Via Gram-Schmidt QR factorization

    • 4.4.3 Via the Householder QR factorization

    • 4.4.4 \(A \) has linearly dependent columns

  • 4.5 Enrichments

    • 4.5.1 Rank-Revealing QR (RRQR) via MGS

    • 4.5.2 Rank revealing Householder QR factorization

  • 4.6 Wrap Up

    • 4.6.1 Additional homework

    • 4.6.2 Summary