The dimension of the null space of a matrix can be inferred from which property?

The dimension of the null space of a matrix is fundamentally linked to the rank-nullity theorem. This theorem states that for any matrix ( A ), the sum of the rank of the matrix (the dimension of the column space) and the nullity of the matrix (the dimension of the null space) is equal to the number of columns in the matrix.

Formally, the theorem can be expressed as:

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

where ( n ) is the total number of columns in the matrix ( A ). From this relationship, if the rank is known, one can easily determine the nullity, and thus the dimension of the null space.

Understanding this theorem helps facilitate calculations regarding the structure of linear transformations represented by the matrix and aids in analyzing the solutions to homogeneous systems associated with the matrix. This is why the rank-nullity theorem is essential for determining the dimension of the null space.

The other options do not provide a direct route to understanding the dimension of the null space. The number of pivot columns relates to the rank but does not directly account for the nullity. The determinant is associated with the matrix's invertibility and does not provide