What is the relationship between the null space and column space of a matrix?

The null space and column space of a matrix are fundamental concepts in linear algebra that highlight different aspects of the matrix's properties and behavior.

The null space, also known as the kernel of the matrix, consists of all vectors that, when the matrix is multiplied by them, yield the zero vector. This directly correlates to linear dependencies among the rows of the matrix, as a non-trivial solution (other than the zero vector) exists if the rows are linearly dependent. Thus, the null space provides insight into the solutions to the homogeneous system of linear equations described by the matrix.

On the other hand, the column space, or range of the matrix, is the set of all linear combinations of the column vectors of the matrix. It represents all possible outputs of the linear transformation defined by the matrix when applied to input vectors, indicating the dimensions and the span of the matrix’s column vectors.

Understanding this relationship clarifies that while the null space deals with the solutions to the equation ( Ax = 0 ), reflecting dependencies among the rows, the column space concerns itself with the outputs and spans of the column vectors pertaining to the matrix. This distinction is crucial for grasping the concepts of linear transformations, independence, and dimensions in vector spaces.