The matrix-specific definition is a special case of this abstract definition.

Height is also used as a name for some more abstract definitions.

In such cases, the concept of direction loses its ordinary meaning, and is given an abstract definition.

On a theoretical level, refinements to the abstract definition of scale-free have been proposed.

It's only during the early 20th century that the current abstract definition of a group without any reference to group actions was used instead.

Since this is a very abstract definition, an illustration will be helpful.

Before giving the abstract definition we give a few examples:

We can solve both problems using a more abstract definition of the exponential map that works for all Lie groups, as follows.

This leads to the more general abstract definition.

This abstract definition coincides with the one in linear algebra.