Rank–Nullity Theorem
Rank plus nullity equals number of columns.
Key idea
Rank plus nullity equals number of columns.
In depth
For an m × n matrix, rank + nullity = n: every input dimension either survives (rank) or gets lost to the null space. It is the conservation law of linear algebra.
Worked example
A 3 × 5 matrix with rank 2 has nullity 3.
Formula
Real-world uses
- Checking the number of free variables in a linear model before solving it
Common mistake
Using m instead of n.
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 Rank–Nullity Theorem?
Rank plus nullity equals number of columns. It is a topic at the expert level in Linear Algebra.
What is the formula for Rank–Nullity Theorem?
The key formula is rank + nullity = n.
Where is Rank–Nullity Theorem used in real life?
Real-world examples: Checking the number of free variables in a linear model before solving it.
What should I learn before Rank–Nullity Theorem?
Make sure you are comfortable with Rank, the previous topic in Linear Algebra.
What is a common mistake with Rank–Nullity Theorem?
Using m instead of n.
How can I practise Rank–Nullity Theorem?
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 Rank–Nullity Theorem?
Move on to Four Fundamental Subspaces 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.