Dumb question:

 

Is the infinite dimensional vector space (with finite number of non-zero components of course) over a countable field countable?

 

I tried to look it up, but couldn't find anything.

 

I don't need a proof, just maybe a little plausible explanation if possible.

Not necessarily, since you've not said if there is a countable basis. There is no such thing as "the" infinite dimensional vector space. You are asking is

 

[math]\coprod_{a \in \Alpha} F_a[/math]

 

countable if each F_a is countable. That depends on the indexing set \Alpha