According to the Rank-Nullity Theorem, what is true about a matrix's rank and nullity?

The Rank-Nullity Theorem is a fundamental result in linear algebra that relates the dimensions of a vector space and its linear transformations, specifically concerning a matrix. According to this theorem, for any matrix ( A ) of size ( m \times n ), the rank of the matrix (the dimension of the image of the corresponding linear transformation) plus the nullity of the matrix (the dimension of the kernel or null space) equals the number of columns ( n ) of the matrix.

This relationship can be expressed mathematically as:

[ \text{Rank}(A) + \text{Nullity}(A) = n ]

Therefore, if a correct statement reflects this principle, stating that the rank plus the nullity equals the number of columns is a precise interpretation. In simpler terms, the rank represents how many dimensions are accounted for in the output of the matrix (or the image), while the nullity counts the dimensions that are "lost" or do not contribute to the output, resulting from the solution to the homogeneous equation ( Ax = 0 ).

Understanding this relationship clarifies why the assertion about the rank and nullity summing to the number of columns is indeed the correct choice. In contrast,