* There's almost always a simple geometric intuition, and low-dimensional intuition can get you quite far even in high dimensional cases.
* You can surprisingly often get by with closing your eyes and saying "my problem is linear" three times. See: All of neural networks.
* Linear problems have practically all nice properties you could ever ask of any function.
Has made linear algebra by far the most bang/buck mathematics topic I've studied in my life. Close behind is asymptotic analysis.
