Four Fundamental Subspaces
Column, row, null and left-null spaces.
Key idea
Column, row, null and left-null spaces.
In depth
For A (m × n): column space and left null space live in Rᵐ, row space and null space live in Rⁿ. Row space ⟂ null space, and column space ⟂ left null space. Together they fully describe the action of A.
Worked example
For rank r: dimensions are r, r, n − r, m − r.
Real-world uses
- Understanding solvability of least squares problems in engineering and data analysis
Common mistake
Mixing up which subspaces live in which space.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Four Fundamental Subspaces?
Column, row, null and left-null spaces. It is a topic at the expert level in Linear Algebra.
Where is Four Fundamental Subspaces used in real life?
Real-world examples: Understanding solvability of least squares problems in engineering and data analysis.
What should I learn before Four Fundamental Subspaces?
Make sure you are comfortable with Rank–Nullity Theorem, the previous topic in Linear Algebra.
What is a common mistake with Four Fundamental Subspaces?
Mixing up which subspaces live in which space.
How can I practise Four Fundamental Subspaces?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Four Fundamental Subspaces?
Move on to Inner Product Spaces next.
Related expert topics in Linear Algebra
- Triple Product – Scalar triple product gives parallelepiped volume.
- Homogeneous Systems – Ax = 0 always has the trivial solution.
- Cofactor Expansion – Compute determinants recursively.
- Properties of Determinants – Multiplicative; row swaps change sign.
- Geometric Meaning of Determinants – Area or volume scaling factor.
- Cramer’s Rule for Systems – Solve systems using determinants.