Rank and Nullity Finder
Determine the dimension of the Column Space and Null Space.
Rank (r)
Nullity (k)
Theorem Verification
0 + 0 = 0 (Cols)
Column Space Visualizer
Visual: The matrix is reduced to RREF.
Green columns have “leading 1s”. Red columns have no pivot.
Understanding Rank and Nullity
Every matrix acts like a machine that transforms vectors. The Rank and Nullity tell us fundamentally how much information this machine preserves and how much it destroys.
Rank (Dimension of Image)
The Rank is the number of linearly independent columns. It tells you the dimension of the output space.
- Rank 2 means the output flattens into a plane.
- Rank 1 means the output collapses onto a line.
Nullity (Dimension of Kernel)
The Nullity is the dimension of the “Null Space”—the set of inputs that get squashed to zero.
It represents the degrees of freedom lost or “hidden” by the transformation.
The Rank-Nullity Theorem
Rank + Nullity = Number of Columns
This theorem is a conservation law for matrices. The number of columns is the dimension of your input space. The theorem says: “The input dimensions must go somewhere—either they appear in the output (Rank) or they get crushed to zero (Nullity).”
Frequently Asked Questions (FAQ)
Q: How do I calculate Rank manually?
A: Perform Gaussian Elimination to get the matrix into Row Echelon Form. Count the number of non-zero rows (or leading 1s). That count is the Rank.
Q: What is “Full Rank”?
A: A matrix has full column rank if Rank = Columns (so Nullity = 0). This means the transformation preserves all dimensions and is one-to-one.
Q: Can Rank be greater than Rows?
A: No. Rank cannot exceed the number of rows OR columns. It is limited by the smaller dimension of the matrix.