What is the rank of the four-by-five matrix example?

Extend a basis of the kernel to a basis of the finite-dimensional domain. The images of the added basis vectors form a basis of the range, giving nullity+rank=domain dimension.

Conditions: The linear map is defined on a finite-dimensional domain.; A basis for the kernel has been identified.

The identity matrix has a two-norm condition number of one. This is because its columns are orthonormal, meaning they are mutually perpendicular and have unit length, which represents the most stable configuration for linear calculations.

Conditions: The matrix is the identity matrix.; The condition number is measured in the two-norm.

Yes, elementary row operations preserve the rank. Reducing a matrix toward echelon form reveals which rows contain pivots and which become zero, allowing the rank to be counted without changing the underlying linear dependencies.

Conditions: The operations are elementary row operations (swapping rows, scaling rows, adding multiples of one row to another).

The rank is equal to the number of pivots in the echelon form. Once the matrix is reduced to echelon form, you simply count the pivot positions; every entry need not be reduced further once pivots are clear.

Conditions: The matrix has been reduced to echelon form.; Pivots are clearly identifiable.

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.