column space and nullspace
(1.3 hours to learn)
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.
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.)
- 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)
→ 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)
→ Multivariable Mathematics
A textbook on linear algebra and multivariable calculus with proofs.
Location: Section 4.4, "The four fundamental subspaces," pages 171-183
- Gaussian elimination
→ 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
- 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
- vector spaces
- linear transformations as matrices
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