# column space and nullspace

(1.3 hours to learn)

## Summary

The column space is the subspace spanned by the columns of a matrix A. The nullspace is the set of solutions to Ax = 0. Both subspaces are useful for characterizing the sets of solutions to linear systems.

## Context

This concept has the prerequisites:

- subspaces (The column space and nullspace are subspaces.)
- linear systems as matrices (The column space and nullspace have to do with the solutions to linear systems.)

## Goals

- Know the definitions of column space and null space

- Show that the column space and null space are subspaces

- Show that Ax = b is solvable iff b is in the column space of A

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

## -Free-

→ Khan Academy: Linear Algebra

→ MIT Open Courseware: Linear Algebra (2011)

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

Location:
Lecture "Column space and nullspace"

## 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

Additional dependencies:

- Gaussian elimination
- bases

→ Introduction to Linear Algebra

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

- Section 3.1, "Spaces of vectors," subsection "The column space of A," pages 123-126
- and Section 3.2, "The nullspace of A: solving Ax = 0," pages 132-138

Additional dependencies:

- Gaussian elimination

→ Linear Algebra Done Right

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

Location:
Chapter 3, "Linear maps," section "Null spaces and ranges," pages 41-47

Additional dependencies:

- bases
- vector spaces
- linear transformations as matrices

## See also

-No Additional Notes-