A column space can have lower dimension than the ambient coordinate space when the columns of the matrix are linearly dependent. In the example, the columns span the xy-plane, which is a two-dimensional subspace of the three-dimensional ambient space, resulting in a matrix rank of two.
Conditions: The matrix has three columns in a three-dimensional ambient space.; The columns are linearly dependent and confined to a plane.
A column space can have lower dimension than the ambient coordinate space when the columns of the matrix are linearly dependent. In the example, the columns span the xy-plane, which is a two-dimensional subspace of the three-dimensional ambient space, resulting in a matrix rank of two.
Conditions: The matrix has three columns in a three-dimensional ambient space.; The columns are linearly dependent and confined to a plane.