# projection onto a subspace

(2.3 hours to learn)

## Summary

The projection of a vector b onto a subspace X is the closest point to b contained in X. Projection is a linear operation, and can be computed using a projection matrix. It is used in linear least squares approximation.

## Context

This concept has the prerequisites:

- subspaces (The projection is onto a subspace.)
- bases (Applying the projection formula requires finding a basis.)
- orthogonal subspaces (The error vector is perpendicular to the subspace.)
- matrix transpose (The matrix transpose is used in the formula for the projection matrix.)
- matrix inverse (The matrix inverse is used in the formula for the projection matrix.)

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

## -Free-

→ Khan Academy: Linear Algebra

- Lecture "Projections onto subspaces"
- Lecture "Visualizing a projection onto a plane"
- Lecture "A projection onto a subspace is a linear transformation"
- Lecture "Subspace projection matrix example"
- Lecture "Another example of a projection matrix"
- Lecture "Projection is the closest vector in a subspace"

→ MIT Open Courseware: Linear Algebra (2011)

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

## -Paid-

→ Introduction to Linear Algebra

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

Location:
Section 4.2, "Projections," pages 206-212

## Supplemental resources (the following are optional, but you may find them useful)

## -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 "Data fitting," pages 225-230

Additional dependencies:

- Lagrange multipliers

## See also

-No Additional Notes-