What does the rank-nullity theorem state?

The rank-nullity theorem is a fundamental result in linear algebra that relates the dimensions of a linear transformation's domain to its range and kernel. Specifically, the theorem states that for a linear transformation represented by a matrix, the sum of the rank (the dimension of the image of the transformation) and the nullity (the dimension of the kernel, or the null space) equals the number of columns of the matrix.

This relationship can be expressed mathematically as:

[ \text{Rank} + \text{Nullity} = n ]

where ( n ) is the number of columns in the matrix. This means that if you know the rank of a matrix, you can easily find the nullity and vice versa, as long as you know the total number of columns. The implications of this theorem are crucial in understanding the properties of linear transformations and the solutions of associated linear systems.

In contrast, the other statements do not accurately reflect the essence of the rank-nullity theorem. The claim about rank being equal to nullity is a misunderstanding of the relationship; while they are related, they are not equal. Saying that the rank is always less than the rank of its inverse does not express a valid property under the theorem's context.