Finding Rank and Null Space Using MATLAB

The rank tells you how much information a matrix contains, while the null space reveals what gets "lost" in the transformation.

In this guide, we'll use MATLAB to compute both properties and verify one of linear algebra's most important theorems.

What are Rank and Null Space?

Rank

The rank of a matrix is the number of linearly independent columns (or rows). It measures the dimension of the column space.

Null Space

The null space (or kernel) of a matrix A is the set of all vectors x such that:

Ax = 0

The dimension of the null space is called the nullity.

The Rank-Nullity Theorem

For any m×n matrix A:

rank(A) + nullity(A) = n

where n is the number of columns.

Step 1: Enter the Matrix

C = [-1 2 0 4 5 -3;
      3 -7 2 0 1 4;
      2 -5 2 4 6 1;
      4 -9 2 -4 -4 7];

This is a 4×6 matrix.

Step 2: Compute the Rank

MATLAB has a built-in function:

rankC = rank(C)

Result: The rank is 2.

This means only 2 columns are linearly independent.

Step 3: Compute the Null Space

nullbasisC = null(C,'r')

The 'r' flag tells MATLAB to use the rational basis (exact fractions instead of decimals).

Result: MATLAB returns a 6×4 matrix where each column is a basis vector for the null space.

Step 4: Interpret the Null Space

If the null space has 4 basis vectors, the nullity is 4.

Each column of nullbasisC is a solution to Cx = 0.

You can verify this:

C * nullbasisC(:,1)  % Should be approximately [0; 0; 0; 0]

Step 5: Verify the Rank-Nullity Theorem

rank(C) + size(nullbasisC, 2)

Result:

• Rank = 2
• Nullity = 4
• Total = 6 (number of columns) ✓

The Rank-Nullity Theorem is satisfied!

Understanding the Null Space Geometrically

The null space represents all the ways you can combine the columns of C to get the zero vector.

If the nullity is large, it means:

• Many columns are redundant
• The transformation "collapses" a lot of information

Alternative: Using RREF

You can also find the null space manually using RREF:

rref(C)

• Pivot columns → correspond to basic variables
• Non-pivot columns → correspond to free variables

Each free variable gives you one basis vector for the null space.

Quick Reference

Key Takeaways

1. Rank = number of pivot columns in the RREF
2. Null space vectors correspond to free variables
3. Rank + nullity always equals the number of columns

Understanding these concepts is crucial for solving systems of equations, analyzing transformations, and working with subspaces!