I will point out that I have taken liberty of creating tag-info which specifies that this tag is for the meaning of the word basis used in linear algebra. I have also mentioned in the tag excerpt the tags schauder-basis and groebner-basis.
If the users will follow the usage suggested in the tag-wiki and tag-excerpt (which more experienced users tend to do), then this should deal with ambiguity of the tag name. And if the tag stays, we can add a new template to the comment templates for tags that are often used incorrectly.
If you consider the tag ambiguous even despite the above suggestions, we can still create a new tag with more explicit name, for example basis-vector-space, and create a synonym with the better named tag as a master tag.
I do not have any strong preference, whether the basis tag should be removed or whether it should stay. But my preferred solution if the tag should be removed would be make it synonym of (linear-algebra) (with linear-algebra as a master tag). This prevents users from creating the tag with the same name again and retagging the questions currently tagged basis would not be needed - with the exception of the few questions where this tag is used for basis of a topological space. (BTW I have already started retagging those questions.)
At the moment there are two suggested synonyms for linear-algebra, namely basis and span. The latter was previously discussed here.