What does it indicate if a matrix has a rank less than its number of columns?

When a matrix has a rank that is less than the number of its columns, it indicates specific properties related to the linear relationships between its columns.

The rank of a matrix is defined as the dimension of its column space, which reflects the maximum number of linearly independent columns in the matrix. If the rank is less than the number of columns, it means that there are not enough linearly independent columns to account for every column in the matrix. This establishes that some of the columns can be expressed as linear combinations of others, indicating that they are linearly dependent.

Additionally, having a rank that is less than the number of columns means the matrix does not span the entire column space. In a system of linear equations represented by this matrix, there will be some solutions that result in infinite solutions because of the inability to constrain all variables independently. This corresponds to the existence of a non-empty null space, meaning there is at least one non-trivial solution to the homogeneous equation associated with the matrix.

Thus, the correct interpretation is that both the linear dependence of the columns and the existence of a non-empty null space are inherent properties of the matrix when its rank is less than its number of columns. This rationale confirms that both statements regarding linear dependence and