What is rank nullity theorem?

The Rank-Nullity Theorem, also known as the Dimension Theorem, is a fundamental result in linear algebra that relates the rank and nullity of a matrix.

The theorem states that for any matrix A, the sum of the rank and nullity of A is equal to the number of columns of A. In other words, if A is an m x n matrix, then rank(A) + nullity(A) = n.

Here, the rank of a matrix is defined as the maximum number of linearly independent columns in the matrix, while the nullity is the dimension of the null space (or kernel) of the matrix, which is the set of all solutions to the equation Ax = 0.

The Rank-Nullity Theorem has important implications in many areas of mathematics and science, including linear transformations, systems of linear equations, and eigenvalues. It is used to determine the dimension of the range and null space of a matrix, as well as to establish the existence of solutions to certain linear systems.