The rank of a matrix is a fundamental concept in linear algebra that describes the dimension of the vector space generated by its columns, which is called the column space of the matrix. Alternatively, it can also be defined as the dimension of the vector space generated by its rows, known as the row space. These two definitions are equivalent, and the rank provides key insights into the matrix's properties and the system of equations it represents.

Key Interpretations of Rank

• Number of Linearly Independent Rows or Columns:
  • Row Rank: The rank of a matrix is defined as the maximum number of rows that are linearly independent. Linear independence implies that no row can be expressed as a linear combination of the other rows.
  • Column Rank: Similarly, the rank of a matrix can also be defined as the maximum number of columns that are linearly independent.
  • Importantly, the row rank and column rank of a matrix are always equal, confirming a fundamental theorem in linear algebra.
• Dimension of the Image (or Column Space): The rank indicates the dimension of the output space into which the matrix, as a linear transformation, maps.
• Dimension of the Row Space: The rank defines the dimension of the vector space spanned by the rows of the matrix.

How to Find the Rank

• Row Echelon Form: Convert the matrix to row echelon form or reduced row echelon form. The number of nonzero rows in this form is the rank of the matrix.
• Gaussian Elimination: This method involves simplifying the matrix into row echelon form to identify the linearly independent rows.
• Determinants and Minors: For small matrices, check if the determinant is nonzero, indicating full rank. For larger matrices, identify the highest order nonzero minor.

Why Does Rank Matter?

• Solving Systems of Equations: The rank helps determine the solvability and type of solutions (none, one, or infinite) for a system of linear equations.
• Invertibility: A square matrix is invertible if it has full rank, equating to the number of its rows or columns.
• Data Representation: In data science, the rank reveals the amount of redundancy within a dataset represented by a matrix.

Example

Consider the matrix:

[ 1  2  1 ]
[ 2  4  2 ]
[ 3  6  3 ]

This matrix has a rank of 1 because all rows and columns are multiples of each other. Geometrically, this represents a transformation that collapses a three-dimensional space onto a single line.

Advanced Note: Rank and Dimension Theorems

• Rank-Nullity Theorem: This theorem states that the sum of the rank of a matrix and the dimension of its null space (kernel) equals the number of columns of the matrix. It provides essential insights into the structure of linear mappings represented by matrices.

Further Readings

• Rank