Matrix Rank, Null Space, and Rank-Nullity

Vector Spaces Associated with a Matrix

Every matrix A of size m x n defines a linear transformation that maps vectors from Rⁿ to R^m. Associated with this mapping are three fundamental vector subspaces:

• Row Space: The span of the rows of A, residing in Rⁿ.
• Column Space: The span of the columns of A, residing in R^m. The column space represents all possible outputs (image) of the transformation.
• Null Space (Kernel): The set of all input vectors v in Rⁿ that map to the zero vector: Av = 0.

What is Matrix Rank?

The dimension of the row space and the column space are mathematically identical. This shared dimension is called the rank of the matrix, denoted as rank(A). The rank tells us the number of dimensions in the output space of the transformation. To find the rank by hand:

1. Perform Gaussian elimination to reduce the matrix to Row Echelon Form (REF) or RREF.
2. Count the number of non-zero rows (rows containing a pivot).
3. This count is the rank of the matrix.

What is the Null Space?

The null space consists of all vectors that are mapped to zero. The dimension of the null space is called the nullity of the matrix. To find a basis for the null space:

1. Perform Gauss-Jordan elimination on the matrix to reduce it to RREF.
2. Write out the equations representing the homogeneous system Av = 0.
3. Identify the free variables (columns without pivots).
4. Express the pivot variables in terms of the free variables, and pull out the coefficients of the free variables to obtain the basis vectors.

The Rank-Nullity Theorem

The Rank-Nullity Theorem is one of the most important theorems in linear algebra. It establishes a conservation law for the dimensions of vector spaces. For any matrix A with n columns:

rank(A) + nullity(A) = n

In other words, the number of independent pivot columns (rank) plus the number of independent free variable columns (nullity) must equal the total number of columns in the matrix.

Worked Example

Consider a 3x3 matrix A whose RREF is:

We analyze the RREF matrix:

• There are 2 non-zero rows. Therefore, rank(A) = 2.
• The columns with pivots are Column 1 and Column 3. Column 2 does not have a pivot, meaning v₂ is a free variable.
• The homogeneous equations are v₁ + 2v₂ = 0 and v₃ = 0.
• Writing the solution vector v = [v₁, v₂, v₃]^T in terms of free variable v₂ yields: v = [-2v₂, v₂, 0]^T = v₂ × [-2, 1, 0]^T.
• Therefore, the null space basis is {[-2, 1, 0]^T}, and nullity(A) = 1.
• Applying the theorem: rank(A) + nullity(A) = 2 + 1 = 3 (number of columns). The theorem holds.