Look at your options within UCLA for music participation as both a major and non-major. Depending on your level, experience, and instrument, you may find programs or options that will allow you to participate within the department (check music dept webpage, music undergrad handbook for policies). Instuction with faculty is dictated by school policy.
It may behoove you to consider private instruction outside of UCLA while pursuing academics.
The discussion here brings up many views.
Joint programs????
Consider the points made on all sides, and choose the plan that suits your current needs and goals.
There is no magic plan for all.
Good luck.