I’m also a one year transfer, and I got into UCSB for Mathematical Sciences just recently. So based on my experience, it is definitely possible for you to transfer in one year! HOWEVER, it’s going to take a lot of effort! I only had 24 units from 4 AP’s (Chinese, Physics Mechanics, US History, Calc AB) and I’m working my butt off in freaking Multivariable Calculus, Differential Equations, and Linear Algebra… I didn’t even get IGETC done, so I feel extremely lucky to get accepted.
Anyway, to find out how many credits you’ll have, go here: http://admission.universityofcalifornia.edu/counselors/files/ap-satisfy-admission-and-igetc-req.pdf
You need to figure out your major ASAP, and finish taking ALL of the course prereqs since the completion of prereqs is something that UCSB is super strict about. Also, do IGETC, which are general education classes, you can research more about it online, but remember major prereqs should be your priority, especially for a one-year transfer! Start out by taking English and Math because those are courses you need at minimum to even transfer. These are the minimum UC transfer requirements:
Two transferable college courses (3 semester or 4-5 quarter units each) in English composition
One transferable college course (3 semester or 4-5 quarter units) in mathematical concepts and quantitative reasoning;
Four transferable college courses (3 semester or 4-5 quarter units each) chosen from at least two of the following subject areas:
the arts and humanities
the social and behavioral sciences
the physical and biological sciences
You can also ask me if you have any questions! Good luck!