# linear least squares

(1.9 hours to learn)

## Summary

Linear least squares gives a value of x which minimizes the norm of Ax - b. It is well defined even in cases where Ax = b has no solution. It is the basis of linear regression, one of the most widely used methods in statistics.

## Context

This concept has the prerequisites:

- linear systems as matrices (Linear least squares generalizes solving systems of linear equations.)
- projection onto a subspace (Projection is used in the solution.)
- matrix transpose (The matrix transpose is used in the normal equations.)
- partial derivatives (The solution can be derived using partial derivatives.)
- four fundamental subspaces (The solution can be derived geometrically in terms of column spaces and nullspaces.)

## Core resources (read/watch one of the following)

## -Free-

→ Khan Academy: Linear Algebra

- Lecture "Least squares approximation"
- Lecture "Least squares examples"
- Lecture "Another least squares example"

→ MIT Open Courseware: Linear Algebra (2011)

Videos for an introductory linear algebra course focusing on numerical methods.

## -Paid-

→ Multivariable Mathematics

A textbook on linear algebra and multivariable calculus with proofs.

Location:
Section 5.5, "Projections, least squares, and inner product spaces," up to "Orthogonal bases," pages 225-232

Additional dependencies:

- Lagrange multipliers

→ Introduction to Linear Algebra

An introductory linear algebra textbook with an emphasis on numerical methods.

Location:
Section 4.3, "Least squares approximation," pages 218-225

## See also

- Linear least-squares is the basis for linear regression , a widely used statistical model.
- Some ways of solving the linear least-squares problem include:
- the QR decomposition
- the pseudoinverse