# Linear map

Thus, a linear map is said to be *operation preserving*. In other words, it does not matter whether the linear map is applied before (the right hand sides of the above examples) or after (the left hand sides of the examples) the operations of addition and scalar multiplication.

For a linear operator with finite-dimensional kernel and co-kernel, one may define *index* as:

No classification of linear maps could be exhaustive. The following incomplete list enumerates some important classifications that do not require any additional structure on the vector space.