
Linear algebra is very powerful.

And there are many “things” in math that behave like vectors.
 Like functions:
 Functions can be added up and scaled similar to vectors. It’s hard to define the exact effect, but basically the result of the summed function is the same as summing the individual results of each of the functions.
 Like functions:

So a vector space is a basically a set of “things”, with constraints defined by the creator of the vector space, such that the set follows the basic axioms of vector addition and scaling.

The axioms are basically a checklist such for the space to be considered a vector space.

This is useful because this enables us to use the tools of linear algebra anywhere we have “vectorish” things.

It’s nice to learn about these tools using regular vectors as it aids visualization and gives us a geometric meaning, but these tools are useful even if our “things” are not really vectors.