# orthogonal subspaces

(1.6 hours to learn)

## Summary

Two subspaces X and Y are orthogonal if any vector in X is orthogonal to any vector in Y. Canonical examples include the row space and nullspace, and the column space and left nullspace. Orthogonality is used to analyze projections and least squares approximation.

## Context

This concept has the prerequisites:

- subspaces
- dot product (The dot product is used to define orthogonality)
- four fundamental subspaces (The four fundamental subspaces include the canonical examples of pairs of orthogonal subspaces.)

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

## -Free-

→ MIT Open Courseware: Linear Algebra (2011)

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

→ Khan Academy: Linear Algebra

## -Paid-

→ Introduction to Linear Algebra

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

Location:
Section 4.1, "Orthogonality of the four subspaces," pages 195-200

## 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 4.4, "The four fundamental subspaces," pages 171-183

## See also

-No Additional Notes-