Vector space

So, in summary, finite-dimensional linear algebra may be expressed in three equivalent languages:

The first example above reduces to this example, if an arrow is represented by a pair of Cartesian coordinates of its endpoint.

and the second and third isomorphism theorem can be formulated and proven in a way very similar to the corresponding statements for groups.

Therefore, the needs of functional analysis require considering additional structures.
The succeeding snapshots show summation of 1 to 5 terms in approximating a periodic function (blue) by finite sum of sine functions (red).