Look in the University/School catalog and see what the prereq/coreqs are first for that course. Then check your current syllabus and see if it mentions that Lin Alg is a prereq. or just recommended. If neither is true then you have some good arguments.
Other considerations: understand that prereqs can be waived if demonstrated knowledge of that material. Waiving prereqs is entirely an administrative issue. If your a Math Major Consult your department dean. If not make sure you consult your respective deans office.
Be clear that you are not asking them to waive the prereq so you do not have to take the course later. Yet that you just want to finish your current course. Make sure that he/she or anyone involved understands that you are NOT trying to waive a Graduation requirement/and or course and still wish to take this course next semester.
If dropping this course would advesely affect your financial aid make sure you show that to you advisor/dean as well. This could be a major issue possibly. So check with financial aid before you let them drop anything!
My University allows Calc 3 as a coreq (not
Required but recommended) with Lin Alg and intro to higher math...talk about too many math classes at one time!
Of course as always having the instructors blessings goes a long way as well.
Always do the leg work for the chair. Go in having all of his questions answered that you possibly can with as many recommendations and signatures required. They want to see action and determination not someone waiting for them to solve their problem. I know this personally! (which your not doing, your trying to go in informed just posting here).
If all else fails and this will significantly affect your academic career goals and/or graduation year/semester. See if your school has a student affairs appeal/grievance office then use it!
I doubt it very seriously your situation going that far and I would not tell anyone you talk to that you are going to do so, just use it as a last resort if you have no other choice!
Above all be respectful, and if waiting to take the class will not affect your financial aid or your academic goal or graduation date, then my advice is to take into serious consideration of dropping the course. Conditional that when to do they place it as an incomplete and take it during the summer, or make sure they back-date the withdraw date so you do not end up with a "W" on your transcript or can get a full refund. Keep in mind your advisor and the department should have caught this during the first week too, but if you knew this to be the case up front too it's just as much your fault as theirs.
Unfortunately I know most of this by experience. I have had several prereqs waived for courses and been able to register for the impossible. Just remember that prereqs being waived is a complete administrative issue and people who can help you are far more willing to help you if you do the work for them!
Lots of information I know but I have been there done that.Maybe I have just been lucky but I have shaved two years off my expected graduation requirements do to prereq waival, hard work, and lots of foot work.
Let us know how it turns out!