The rank of a matrix is the dimension of its column space: how many independent directions survive the transformation.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Counting it
Rank equals the number of pivots after row reduction. Each pivot marks one genuinely new direction.
Nullity
The nullity is the dimension of the null space: how many independent directions get crushed to zero.
The theorem
Rank plus nullity equals the number of columns. Every input direction is either preserved or destroyed, and nothing is left over.
Reading it
A high rank means little was lost. A high nullity means the transformation flattened a great deal.
Full rank
A square matrix of full rank has nullity zero, which makes it invertible. The theorem and the invertibility theorem agree.
Rank is the surviving dimension
The rank of a matrix is the dimension of its column space — how many dimensions come through the transformation intact. It equals the number of pivots after elimination.
Nullity is what gets crushed
The null space contains everything sent to zero, and its dimension is the nullity. It counts the directions the transformation collapses, and it equals the number of free variables.
The theorem
Rank plus nullity equals the number of columns. What survives plus what is crushed accounts for the whole input space, which makes the theorem almost a bookkeeping identity once stated that way.
What it tells you immediately
Full rank means nothing collapses, so the null space is trivial and the transformation is injective. Low rank means substantial collapse, which is exactly what data compression exploits.
Step 2: Try It Yourself
Tap and try it out.
- Determinant of A-1
A non-zero determinant means the matrix is invertible. Its size is the factor by which areas are scaled — here 1.
Step 3: Watch an Example
One step at a time.
Watch Kwame Balance the Books
Kwame has a 3 by 5 matrix of rank 3 and wants its nullity.
- Step 1
He counts the columns, which is 5, and that is the total to be accounted for.
Step 4: Your Turn
Practice makes it stick.
The Balance
Problem 1 of 2
A matrix with 7 columns and rank 4. What is its nullity?
The Full One
Problem 2 of 2
An invertible 5 by 5 matrix. What is its nullity?
Balance the Dimensions
1 of 8
6 columns and rank 2. What is the nullity?
2 of 8
4 columns and nullity 1. What is the rank?
3 of 8
A 2 by 2 matrix with determinant 0 and non-zero entries. What is its rank?
4 of 8
The zero matrix with 3 columns. What is its rank?
5 of 8
The zero matrix with 3 columns. What is its nullity?
6 of 8
A 3 by 3 identity matrix. What is its rank?
7 of 8
Match each quantity to what it measures.
Tap a card on the left to start.
8 of 8
10 columns and rank 10. What is the nullity?
Step 5: Quick Check
Show what you know.
Question 1 of 2
8 columns and rank 5. What is the nullity?
Question 2 of 2
What does the rank-nullity theorem say?
What You Learned
- Rank is how many independent directions survive; nullity is how many are crushed.
- Rank plus nullity always equals the number of columns.
- A square matrix of full rank has nullity zero and is therefore invertible.