Rank
The dimension of the column space.
Key idea
The dimension of the column space.
In depth
The rank is the number of pivots, the dimension of the column space, which equals that of the row space. It counts the independent directions in the output of the transformation.
Worked example
A = [1 2; 2 4; 3 6] has rank 1.
Formula
Real-world uses
- Measuring how many independent measurements a dataset contains
- Checking if a model is solvable
Common mistake
Counting non-zero rows before fully reducing.
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?
The dimension of the column space. It is a topic at the expert level in Linear Algebra.
What is the formula for Rank?
The key formula is rank(A).
Where is Rank used in real life?
Real-world examples: Measuring how many independent measurements a dataset contains; Checking if a model is solvable.
What should I learn before Rank?
Make sure you are comfortable with Null Space, the previous topic in Linear Algebra.
What is a common mistake with Rank?
Counting non-zero rows before fully reducing.
How can I practise Rank?
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?
Move on to Rank–Nullity Theorem 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.