Rank Nullity Theorem: Proof & Intuition (Linear Algebra)

For an m×n matrix A, the rank plus nullity equals the number of columns n: rank(A) + null(A) = n. This theorem is fundamental because knowing the rank automatically determines the nullity. Additionally, the rank of a matrix equals the rank of its transpose. A matrix is invertible (non-singular) if and only if its rank equals its number of columns (for square matrices), or equivalently, if its nullity is zero. These relationships allow determining matrix properties without full computation.

For any matrix A with m columns, rank(A) + nullity(A) = m. This follows because each column either contains a leading one (contributing to rank) or does not (contributing to nullity). For an n×n matrix A, the following conditions are equivalent: rank(A) = n, nullity(A) = 0, A is invertible, Ax = b has a unique solution for every b, the column vectors form a basis for R^n, the rows form a basis for R^n, and det(A) ≠ 0.

The Invertible Matrix Theorem states that for an n×n matrix, twelve specific conditions are all equivalent—meaning they are either all true or all false together. These conditions include: the matrix being invertible, having n pivot positions, being row equivalent to the identity matrix, having only the trivial solution to Ax=0, having linearly independent columns, being one-to-one and onto as a linear transformation, having columns that span R^n, and having both a left and right inverse. This theorem provides a comprehensive framework for determining whether a square matrix is invertible by checking any one of these equivalent conditions.

The rank-nullity theorem states that for any m×n matrix A, rank(A) + nullity(A) = n, where n is the number of columns. This fundamental relationship allows calculation of one quantity when the other is known. For an n×n matrix, det(A) ≠ 0 implies nullity(A) = 0 (invertible matrix), while non-zero nullity implies det(A) = 0 and rank < n. Scalar matrices (diagonal with equal non-zero entries) have nullity 0. For non-zero diagonal matrices of order 3, nullity depends on zero entries: 0, 1, or 2. Nullity of cA equals nullity of A for any non-zero scalar c. The nullity of the sum of two matrices is not necessarily equal to the sum of their nullities—a counterexample is A = I and B = -I, where A + B = 0 has nullity n while both A and B have nullity 0.

The nullity of a matrix is the number of zero columns. For a square matrix of order n×n, nullity = n - Rank(A). For a rectangular matrix of order m×n, nullity = n - Rank(A), where n is the number of columns. The nullity represents the dimension of the null space. A matrix is invertible if and only if its nullity is zero. This theorem connects rank and nullity for any matrix.

The four fundamental subspaces of an m×n matrix A are the row space (subspace of R^n spanned by the rows of A), the column space (subspace of R^m spanned by the columns of A), the null space (solution space of Ax=0 in R^n), and the left null space (null space of A^T, also called the null space of A transpose, which is a subspace of R^m). These subspaces satisfy two key relationships: (1) the row space and null space are orthogonal complements in R^n, and (2) the column space and left null space are orthogonal complements in R^m. The dimensions of these subspaces can be determined from the size of the matrix and its rank r: dim(row space) = r, dim(column space) = r, dim(null space) = n - r, and dim(left null space) = m - r. This collection of results is known as the fundamental theorem of linear algebra.

The fundamental theorem of linear algebra states that for any m×n matrix A with rank r, the four fundamental subspaces—column space (R(A)), null space (N(A)), row space (R(A^T)), and left null space (N(A^T))—are related by dimension formulas: dim(R(A)) = dim(R(A^T)) = r, dim(N(A)) = n - r, and dim(N(A^T)) = m - r, and they satisfy orthogonal complement relationships where R(A) is the orthogonal complement of N(A^T) in R^m and N(A) is the orthogonal complement of R(A^T) in R^n.

A matrix has four fundamental subspaces that are interconnected: (1) Column space - all combinations of the columns, forming a flat surface in n-dimensional space; (2) Row space - all combinations of the rows; (3) Null space (kernel) - perpendicular to the column space, representing solutions to Ax=0; (4) Left null space - perpendicular to the row space. These four spaces fit together in a beautiful picture that reveals the structure of any matrix.

In linear algebra, any matrix defines four fundamental subspaces: the column space (all linear combinations of columns), the row space (all linear combinations of rows), the null space (solutions to Ax=0), and the left null space (solutions to A^T x=0). For a matrix A with m rows and n columns, the dimension of the column space plus the dimension of the null space equals n (the number of columns), and the dimension of the row space plus the dimension of the left null space equals m (the number of rows). These relationships are fundamental to understanding the structure of linear systems and solving equations.

For an m by n matrix A, there are four fundamental subspaces: (1) The range of A (column space) in R^m, (2) The null space of A in R^n, (3) The range of A^T (row space) in R^n, and (4) The null space of A^T in R^m. The fundamental theorem states: (1) The null space of A is the orthogonal complement to the range of A^T, (2) The Cartesian product of the null space of A and the range of A^T equals R^n, and (3) The rank-nullity theorem states that dim(range(A)) + dim(null(A)) = n. The rank of A equals the rank of A^T, representing the number of linearly independent columns or rows.

The rank-nullity theorem helps analyze linear systems: For a matrix with n columns, rank(A) + nullity(A) = n. Nullity equals the number of free variables. When rank(A) < n, there are free variables and infinitely many solutions for Ax = 0. For nonhomogeneous systems Ax = b, compare ranks: if rank(A) < rank([A|b]), no solution exists; if ranks are equal and equal to n, unique solution exists; if ranks are equal but less than n, infinitely many solutions exist. This provides a complete classification of linear system behavior.

This section develops the rank-nullity theorem and its applications to analyzing homogeneous systems. The theorem states that for an m×n matrix A, rank(A) + nullity(A) = n, where n is the number of columns. The nullity is the dimension of the solution space. When rank(A) = n, the solution space has dimension 0, meaning only the trivial solution exists. When rank(A) < n, the solution space has dimension > 0, meaning there are infinitely many solutions. The solution space forms a vector space containing infinitely many elements when non-trivial solutions exist. The section demonstrates this through examples: for [[1,1,1],[1,2,3],[1,1,7]], rank(A) = 2, n = 3, so solution space dimension = 1. For [[1,1,2],[1,2,4],[2,2,4]], rank(A) = 2, n = 3, so solution space dimension = 1.

The Rank-Nullity Theorem establishes that for any m×n matrix A, rank(A) + nullity(A) = n, connecting the dimension of the column space to the dimension of the null space. This theorem provides a complete characterization of solution spaces for homogeneous systems: when rank(A) < n, the null space has dimension n - rank(A) > 0, guaranteeing infinitely many solutions. The theorem also explains why homogeneous systems always have at least the trivial solution and potentially many more when the coefficient matrix is rank-deficient. This fundamental relationship underpins all subsequent analysis of linear system behavior.

For a homogeneous system Ax = 0 with an m×n matrix A, the dimension of the solution space (nullity) is n - rank(A). This is the Rank-Nullity Theorem. The rank represents independent equations, while n is the total number of variables. The difference gives the number of free variables in the general solution.

The rank-nullity theorem has important applications: (1) The dimension of the null space equals the number of linearly independent solutions to Ax = 0, (2) The rank equals the number of linearly independent equations, (3) For Ax = b to have a solution, b must be in the column space of A, (4) If rank equals the number of rows, solutions exist for any b.