four fundamental subspaces
(2.4 hours to learn)
The four fundamental subspaces of a matrix A are the column space, nullspace, row space, and left nullspace. The bases of all four spaces can be obtained using Gaussian elimination, and certain of them are orthogonal to one another. There are close relationships between the dimensions of all four spaces, and the dimensions of the row and column spaces both equal the rank of A.
This concept has the prerequisites:
- column space and nullspace (The column space and nullspace are two of the fundamental subspaces.)
- subspaces (The rank is defined in terms of the dimension of a certain subspace.)
- bases (Analyzing the dimensions of the subspaces requires determining bases for them.)
- Gaussian elimination (Gaussian elimination is used to compute bases for the subspaces.)
- matrix transpose (Different subspaces are related by matrix transposes.)
Core resources (read/watch one of the following)
→ Khan Academy: Linear Algebra
- Lecture "Null space and column space basis"
- Lecture "Visualizing a column space as a plane in R^3"
- Lecture "Dimension of the null space or nullity"
- Lecture "Dimension of the column space or rank"
- Lecture "Showing relation between basis cols and pivot cols"
- Lecture "Showing that the candidate basis does span C(A)"
→ MIT Open Courseware: Linear Algebra (2011)
Videos for an introductory linear algebra course focusing on numerical methods.
Location: Lecture "The four fundamental subspaces"
→ Multivariable Mathematics
A textbook on linear algebra and multivariable calculus with proofs.
Location: Section 4.4, "The four fundamental subspaces," pages 171-183
→ Introduction to Linear Algebra
An introductory linear algebra textbook with an emphasis on numerical methods.
Location: Section 3.6, "Dimensions of the four subspaces," pages 184-189
- In statistics, we often want to approximate a given matrix using a low rank matrix .
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