A injective linear map is any linear map which satisfies the injective property. This property has a few consequences for the matrix which represents the map.

For a map represented by a matrix such that the following properties are all equivalent

1. is a injective linear map
2. For every in the equation has at most one solution.
3. For every in the equation has at most one solution or is inconsistent.
4. has only the trivial solution.
5. The columns of are linearly independent.
6. has a pivot in every column.
7. The range (span of 's columns) has dimensions .