# subspaces

(1.1 hours to learn)

## Summary

A subspace is a subset of a vector space which is itself a vector space. Examples include spans of sets of vectors and solution sets to linear equations of the form Ax = 0.

## Context

This concept has the prerequisites:

- vectors (Subspaces are sets of vectors.)

## 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-

→ Multivariable Mathematics

A textbook on linear algebra and multivariable calculus with proofs.

Location:
Section 1.3, "Subspaces of R^n," pages 16-22

Additional dependencies:

- dot product

→ Introduction to Linear Algebra

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

Location:
Section 3.1, "Spaces of vectors," subsection "Subspaces," pages 121-123

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

## -Free-

→ A First Course in Linear Algebra (2012)

A linear algebra textbook with proofs.

Location:
Section "Subspaces," pages 334-346

Additional dependencies:

- vector spaces
- Gaussian elimination
- complex vectors and matrices

## -Paid-

→ Linear Algebra Done Right

A textbook for a second course in linear algebra, with mathematical generalizations of the basic concepts.

Location:
Chapter 1, "Vector spaces," section "Subspaces," pages 13-14

## See also

- Subspaces are used to define the rank of a matrix , an important concept in solving linear systems.