Hey There,
I was a veteran of the Westside for 5 years having moved to Inglewood from Sacramento. I lived in Inglewood first, then Westchester, then Culver City. Sounds like a lot of places but it was all within a 1 mile radius. I moved back to Sacramento last year.
Biking...I don't suggest it, but you can do it. The new Inosanto Academy is over in the Marina. If you wind up in Santa Monica you can take the bike path from the Pier all the way out to the Marina without much trouble or traffic. There are also busses that run pretty regularly from SMC and UCLA to the Marina. They run right down Lincoln and pass right by the school. The problem for you will be getting back home. The busses get pretty thin from about 8pm on. It also rains in SoCal. Not that often but when it does, the streets aren't friendly to bikes or pedestrians.
Brentwood and Westwood are not close to the beach so that means if you live there then you have to brave some of the most dangerous biking road in the world and run through patches of ghetto to get there. If you don't mind inconvenience, danger, and breathing exhaust, then you don't need a car in LA. But if you want to get places on time and safely, then you need to have one. Otherwise you're at the mercy of the busses, which do not always pick you up, and the masses, which are not always friendly.
SoCal is great, but it is very much a different type of place. I really enjoyed my time there as much as I am glad to be home. I lived at the intersection of La Tijera, Centinela, and La Cienega for most of my time down there and I never had any trouble. LA has slums just a few blocks away from mansions. It's really quite different than what you're used to. My suggestion is keep your car and buy a Thomas Guide. I'd expand your search for apartments or rooms to West LA, especially out by the Santa Monica Airport, Playa Del Rey, and Culver City, instead of Brentwood and the like. Those places are nice but they have a matching price tag.
You should also keep in mind that there is a lot out there besides the Inosanto Academy. You might want to keep your car for additional training with different teachers because there are A LOT down there. Without wheels you are without many options.
Good luck from Your NorCal - SoCal Brother,
Bart Hubbard
Capital Doce Pares